The Limit Set for Representations of Discrete Subgroups of $\text {PU}(1,n)$ by the Plücker Embedding

Abstract

Let $\Gamma$ be a discrete subgroup of $\text {PU}(1,n)$. In this work, we look at the induced action of $\Gamma$ on the projective space $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$ by the Plücker embedding, where $\wedge ^{k+1}$ denotes the exterior power. We define a limit set for this action called the k-Chen-Greenberg limit set, which extends the classical definition of the Chen-Greenberg limit set $L(\Gamma )$, and we show several of its properties. We prove that its Kulkarni limit set is the union taken over all $p\in L(\Gamma )$ of the projective subspace generated by all k-planes that contain p or are contained in $p^{\perp }$ via the Plücker embedding. We also prove a duality between both limit sets.

1 Introduction

The study of the dynamics of discrete group actions on compact manifolds is still in its early developments. For instance, unlike the case of conformal dynamics, there are several possible definitions of the limit set, each having its own properties; see [4] for a thorough discussion.

Consider the subgroup $\text {PU}(r,s)\subset \text {PSL}(n+1,\mathbb {C})$ consisting of all automorphisms of $\mathbb{C}\mathbb{P}^{n}$ that preserve a certain Hermitian form of signature (r, s). Little is known about the dynamics of discrete subgroups of $\text {PU}(r,s)$ in a general way, especially on the existence of limit sets; see [1, 3, 12]. In this article, we consider this question for certain subgroups of $\text {PU}(r,s)$ that arise by considering the action of a discrete subgroup of $\text {PU}(1,n)$ on the Grassmannian of k dimensional subspaces in $\mathbb{C}\mathbb{P}^{n}$ via the Plücker embedding. We will see that these subgroups belong to $\text {PU}(r,s)$, where r and s depend on k and n. We need to recall some facts:

We take $\text {PU}(1,n)\subset \text {PSL}(n+1,\mathbb {C})$, the group of all automorphisms of $\mathbb{C}\mathbb{P}^{n}$ that preserve the unit ball

$$\begin{aligned} \mathbb {H}^{1,n}_{\mathbb {C}}:=\{[z_{0}:z_{1}:\cdots :z_{n}]\in \mathbb{C}\mathbb{P}^{n}: |z_{1}|^{2}+...+|z_{n}|^{2}<|z_{0}|^{2} \}. \end{aligned}$$

This ball can be equipped with the Bergman metric and provides a model for the complex hyperbolic space $\mathbb {H}^{1,n}_{\mathbb {C}}$, with $\text {PU}(1,n)$ as its group of holomorphic isometries; see [6]. In this case, if $\Gamma \subset \text {PU}(1,n)$ is a discrete subgroup and we look at the action of $\Gamma$ on $\mathbb {H}^{1,n}_{\mathbb {C}}$, then one has the limit set defined in the usual way, as the set of accumulation points of the orbit of $\Gamma$ denoted $L(\Gamma )$. This set is a closed $\Gamma$-invariant subset of the boundary of $\mathbb {H}^{1,n}_{\mathbb {C}}$ known as the Chen-Greenberg limit set of $\Gamma$. If this set has finite cardinality, then the group is said to be elementary and its cardinality is at most two, and if the group is non-elementary, then it is a minimal set; see [6].

Also, $\Gamma$ is by definition a subgroup of $\text {PSL}(n+1,\mathbb {C})$ and hence acts on the whole of $\mathbb{C}\mathbb{P}^{n}$, but the action is no longer by isometries. There, the definition of a limit set is not as clear. One option is the region of equicontinuity (see Definition 1.5) of the group $\Gamma$ seen as a set of functions. In [5], the authors proved that the equicontinuity region for $\Gamma \subset \text {PU}(1,n)$, a discrete subgroup, is the complement of the union of all hyperplanes tangent to $\partial \mathbb {H}^{1,n}_{\mathbb {C}}$ at points in $L(\Gamma )$.

On the other hand, the Kulkarni limit set (see Definition 1.7) provides us with an invariant open set where the group acts properly and discontinuously. In [2], the authors determined the Kulkarni limit set of $\Gamma$ as the union of all hyperplanes tangent to $\partial \mathbb {H}^{1,n}_{\mathbb {C}}$ at points in $L(\Gamma )$. This result was proven (essentially) by J. P. Navarrete in [12] for $n = 2$.

We consider the following homomorphism:

$$\begin{aligned} \begin{array}{rccl} \phi _{k}:&{} \text {PSL}(n+1,\mathbb {C})&{}\longrightarrow &{}\mathbb {P} \left( \text{ Aut }(\wedge ^{k+1}\mathbb {C}^{n+1})\right) \\ &{}[\gamma ]&{}\mapsto &{}\left[ \wedge ^{k+1}\gamma \right] . \end{array} \end{aligned}$$

In [16], the author proved that this homomorphism preserves discreteness, i.e., it carries discrete groups into discrete groups. Moreover, she proved that if $\Gamma \subset \text {PU}(1,n)$ is a discrete subgroup, $0\le k< n$, then $\wedge ^{k+1}\Gamma$ is a discrete subgroup of $\text {PU}(R,S)$ acting on $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$, where $R=\left( {\begin{array}{c}n\\ k\end{array}}\right)$ and $S=\left( {\begin{array}{c}n\\ k+1\end{array}}\right)$. So, the induced subgroup $\wedge ^{k+1}\Gamma$ preserves a Hermitian form of signature (R, S) and acts on $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$, space isomorphic to the projective space $\mathbb{C}\mathbb{P}^{N-1}$, where $N=R+S$.

Little is known about the dynamics of this kind of discrete subgroup, in particular, about the existence of the largest open sets where the group acts properly and discontinuously. In [16], the author determined its equicontinuity region and proved that the action of $\wedge ^{k+1}\Gamma$ on $Eq(\wedge ^{k+1}\Gamma )$ is properly discontinuously. In this article, we give two candidate limit sets for this kind of induced discrete groups.

We define a limit set called the k-Chen-Greenberg limit set, denoted $L_{k}(\Gamma )$, which extends the classical definition of the Chen-Greenberg limit set to these induced groups. It can be described as the union for all $p\in L(\Gamma )$ of the projective subspace $[\langle H^{k}_{p}\rangle ]$, generated by all k-planes in $\mathbb{C}\mathbb{P}^{n}$ that contain p and are contained in $p^{\perp }$ via the Plücker embedding. This set is a closed $\wedge ^{k+1}\Gamma$-invariant subset of the boundary of $\mathbb {H}^{R,S}_{\mathbb {C}}$. We prove several properties of this limit set; see Theorem 4.9.

This limit set allows us to parameterize the Kulkarni limit set of the group $\wedge ^{k+1}\Gamma$ acting on the projective space $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$ in the following way (it is a generalization of main theorem in [12] by J. P. Navarrete and Theorem 0.1 in [2] by A. Cano et al.):

Theorem 1.1

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k< n$; then, the Kulkarni limit set of $\wedge ^{k+1}\Gamma$ can be described as the union of all projective subspaces orthogonal to the elements of the k-Chen-Greenberg limit set of $\Gamma$, that is,

$$\begin{aligned} \Lambda _{Kul}(\wedge ^{k+1}\Gamma )= \bigcup _{[\langle H^{k}_{p}\rangle ] \subset L_{k}(\Gamma )} \left[ \langle H^{k}_{p}\rangle \right] ^{\perp } . \end{aligned}$$

Moreover, if $\Gamma$ is non-elementary, then $\Omega _{Kul}(\wedge ^{k+1}\Gamma )$ is the largest open set on which $\wedge ^{k+1}\Gamma$ acts properly discontinuously and coincides with $Eq(\wedge ^{k+1}\Gamma )$.

On the other hand, we notice that the k-Chen-Greenberg limit set of $\Gamma$, $L_{k}(\Gamma )$, can be seen as points in $\text {Gr}_{M-1}(\mathbb{C}\mathbb{P}^{N-1})$, where $M=\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$, and the Kulkarni limit set, $\Lambda _{Kul}(\wedge ^{k+1}\Gamma )$, are points in $\text {Gr}_{N-M-1}(\mathbb{C}\mathbb{P}^{N-1})$, Grassmannians that are duals.

To make notation simpler, we take $\widehat{\Gamma }=\wedge ^{k+1}\Gamma$, so the group $\wedge ^{N-M}\widehat{\Gamma }$ acts on the projective space $\mathbb {P}(\wedge ^{N-M}\mathbb {C}^{N})$. We define the limit set of $\widehat{\Gamma }$, $\widehat{L}_{k}(\widehat{\Gamma })$ (see Definition 6.1), as the union for all $p\in L(\Gamma )$ of the projective subspace generated by all $(N-M-1)$-planes that contain $\left[ \langle H^{k}_{p}\rangle \right]$ and are contained in $\left[ \langle H^{k}_{p}\rangle \right] ^{\perp }$ via the Plücker embedding.

For $F \subset \mathbb {P}(\wedge ^{N-M}\mathbb {C}^{N})$ a subset, we denote by $V_{\ell }\subset \mathbb{C}\mathbb{P}^{N-1}$ the complex projective subspace that determines the point $\ell \in F$, then

$$\begin{aligned} \bigcup _{\ell \in F}V_{\ell } \end{aligned}$$

is the subset of $\mathbb{C}\mathbb{P}^{N-1}$ obtained as the union of all $(N-M-1)$-planes determined by the elements in F.

In the following theorem, we prove a duality between the limit set $\widehat{L}_{k} (\widehat{\Gamma })$ of $\wedge ^{N-M}\widehat{\Gamma }$ acting on $\mathbb {P}(\wedge ^{N-M}\mathbb {C}^{N})$ and the Kulkarni limit set $\Lambda _{Kul}(\wedge ^{k+1}\Gamma )$ of $\wedge ^{k+1}\Gamma$ acting on $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$.

Theorem 1.2

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k<n$. Then,

$$\begin{aligned} \Lambda _{Kul}(\wedge ^{k+1}\Gamma )=\bigcup _{\ell \in \widehat{L}_{k} (\widehat{\Gamma })}V_{\ell }. \end{aligned}$$

The paper is organized as follows. In Section 1, we review some general facts and introduce the notation used along with the text. In Section 2, we see the geometry of exterior powers of $\mathbb {C}^{n+1}$. In Section 3, we define the k-Chen-Greenberg limit set. In Section 4, we give a proof of the main Theorem 1.1. In Section 5, we provide a duality between both limit sets.

2 Preliminaries

2.1 Projective Geometry

The complex projective space $\mathbb{C}\mathbb{P}^{n}$ is defined as

$$\begin{aligned} \mathbb{C}\mathbb{P}^{n}=(\mathbb {C}^{n+1}- \{0\})/ \mathbb {C}^{*}, \end{aligned}$$

where $\mathbb {C}^{*}$ acts by the usual scalar multiplication. This is a compact connected complex n-dimensional manifold equipped with the Fubini-Study metric $d_{n}$.

If $[\hspace{2mm} ]: \mathbb {C}^{n+1}- \{0\}\longrightarrow \mathbb{C}\mathbb{P}^{n}$ is the quotient map, then a non-empty set $\text {V}\subset \mathbb{C}\mathbb{P}^{n}$ is said to be a k-dimensional projective subspace if there is a $\mathbb {C}$-linear subspace $\widetilde{\textrm{V}}\subset \mathbb {C}^{n+1}$ of dimension $k+1$ such that $[\widetilde{\textrm{V}}- \{0\}]=\textrm{V}$.

2.2 Projective Transformations

Consider the general linear group $\text{ GL }(n+1,\mathbb {C})$ of invertible square complex matrices with non-zero determinant. It is clear that every linear automorphism of $\mathbb {C}^{n+1}$ defines a holomorphic automorphism of $\mathbb{C}\mathbb{P}^{n}$, and it is well known that every holomorphic automorphism of $\mathbb{C}\mathbb{P}^{n}$ arises in this way. Thus, one has that the group of holomorphic projective automorphisms of $\mathbb{C}\mathbb{P}^{n}$ is as follows:

$$\begin{aligned} \text {PSL}(n+1,\mathbb {C}):=\text{ GL }(n+1,\mathbb {C})/(\mathbb {C}^{*})^{n+1} \cong \text{ SL }(n+1,\mathbb {C})/ Z_{n+1}, \end{aligned}$$

where $(\mathbb {C}^{*})^{n+1}$ is the subgroup of all scalar multiples of the identity, and the action of $Z_{n+1}$ (regarded as the roots of unity) is given by the usual scalar multiplication.

Then, $\text {PSL}(n+1,\mathbb {C})$ is a Lie group whose elements are called projective transformations. We denote by $[\hspace{2mm}]:\text{ SL }(n+1,\mathbb {C})\longrightarrow \text {PSL}(n+1,\mathbb {C})$ the quotient map. Given $\gamma \in \text {PSL}(n+1,\mathbb {C})$, we say that $\tilde{ \gamma } \in \text{ SL }(n+1,\mathbb {C})$ is a lift of $\gamma$ if $[\tilde{\gamma }]=\gamma$.

Notice that $\text {PSL}(n+1,\mathbb {C})$ acts transitively, effectively, and by biholomorphisms on $\mathbb{C}\mathbb{P}^{n}$, taking projective subspaces into projective subspaces.

2.3 The Pseudo-Unitary Space

In what follows, $\mathbb {C}^{r,s}$ is a copy of $\mathbb {C}^{n+1}$ equipped with a Hermitian form $\langle .,.\rangle _{r,s}$, induced by an Hermitian matrix $H_{r,s}$ of signature (r, s), with $r+s=n+1$.

A vector $v\in \mathbb {C}^{r,s} - \{0\}$ is called negative, null, or positive depending on the value of $\langle v,v \rangle _{r,s}$; we denote the set of negative, null, or positive vectors by $V^{r,s}_{-}$, $V^{r,s}_{0}$, and $V^{r,s}_{+}$ respectively. Thus, one has the following:

$$\begin{aligned} \begin{aligned} V^{r,s}_{-}=&\{z \in \mathbb {C}^{r,s} | \langle z,z \rangle _{r,s}<0\},\\ V^{r,s}_{0}=&\{z \in \mathbb {C}^{r,s}-\{0\} | \langle z,z \rangle _{r,s}=0\},\\ V^{r,s}_{+}=&\{z \in \mathbb {C}^{r,s} | \langle z,z \rangle _{r,s}>0\}. \end{aligned} \end{aligned}$$

Let $V\subset \mathbb{C}\mathbb{P}^{n}$ be a proper and non-empty projective subspace, set $\widetilde{V}=\{v\in \mathbb {C}^{n+1} - \{0\}\hspace{1mm}| \hspace{1mm} [v]\in V \}$, and define $V^{\perp }=[\widetilde{V}^{\perp }]$, where $\widetilde{V}^{\perp }$ is the orthogonal complement of $\widetilde{V}$ defined as follows:

$$\begin{aligned} \widetilde{V}^{\perp }= \{w\in \mathbb {C}^{n+1} - \{0\} \hspace{2mm}|\hspace{2mm} \langle v,w \rangle _{r,s}=0 \hspace{2mm} \text {for all} \hspace{2mm} v\in \widetilde{V}\}. \end{aligned}$$

We consider

$$\begin{aligned} \text {U}(r,s)=\left\{ g\in \text{ GL }(n+1,\mathbb {C}): g^{*}H_{r,s}g=H_{r,s} \right\} \end{aligned}$$

where $g^{*}$: denotes the transpose conjugate. So, $\text {U}(r,s)$ is the group of matrices that preserve the Hermitian form $\langle .,.\rangle _ {r,s}$. Without loss of generality, we can assume $r \le s$. Taking the corresponding projectivization, we obtain what $\text{ PU }(r,s)$ preserves the set $\mathbb {H}^{r,s}_{\mathbb {C}}=[V^{r,s}_{-}]$, that is,

$$\begin{aligned} \mathbb {H}^{r,s}_{\mathbb {C}}=\lbrace [w] \in \mathbb{C}\mathbb{P}^{n} \hspace{2mm}\vert \hspace{2mm} \langle w, w \rangle _{r,s}< 0 \rbrace , \end{aligned}$$

whose boundary is $[V^{r,s}_{0}]$ will be denoted by $\partial \mathbb {H}^{r,s}_{\mathbb {C}}$.

For the rest of the article, we are considering the following Hermitian matrix unless otherwise stated:

$$\begin{aligned} H_{1,n}= \begin{pmatrix} &{} &{} 1\\ &{} Id_{n-1} &{} &{} \\ 1 &{} &{} &{} \end{pmatrix} \end{aligned}$$

where $Id_{n-1}$ denotes the identity matrix of $(n-1) \times (n-1)$.

2.4 Classification of the Elements

In [6], Chen and Greenberg classified the elements of $\text {PU}(1,n)$ in the following way:

Definition 2.1

Let $\gamma \in \text {PU}(1,n)$ be a projective transformation, we say that $\gamma$ is as follows:

• (1) Loxodromic if $\gamma$ is diagonalizable with exactly $n-1$ eigenvalues of norm one, and the other two of inverse norms different than one.

• (2) Elliptic if $\gamma$ is diagonalizable with all its unit modulus eigenvalues.

• (3) Parabolic if $\gamma$ is not diagonalizable with all its eigenvalues of norm one.

Subsequently, Goldman [10] gave a classification of the elements of $\text {PU}(1,n)$ based on the amount and location of their fixed points.

Definition 2.2

Let $\gamma \in \text {PU}(1,n)$, then $\gamma$ is as follows:

• (1) Loxodromic if it has an unique pair of points in $\partial \mathbb {H}^{1,n}_{\mathbb {C}}$.

• (2) Elliptic if it has a fixed point in $\mathbb {H}^{1,n}_{\mathbb {C}}$.

• (3) Parabolic if it has an unique fixed point in $\partial \mathbb {H}^{1,n}_{\mathbb {C}}$.

Later, in [3], the authors extended the classification of [6] to every element of $\text {PSL}(n+1,\mathbb {C})$.

Definition 2.3

Let $\gamma \in \text {PSL}(n+1,\mathbb {C})$, then $\gamma$ is as follows:

• (1) Loxodromic if $\gamma$ has a lift $\tilde{\gamma }\in \text {SL}(n+1,\mathbb {C})$ such that $\tilde{\gamma }$ has at least one eigenvalue of norm different from one. It can be diagonalizable or not.

• (2) Elliptic if $\gamma$ has a lift $\tilde{\gamma }\in \text {SL}(n+1,\mathbb {C})$ such that $\tilde{\gamma }$ is diagonalizable and all its eigenvalues of norm one.

• (3) Parabolic if $\gamma$ has a lift $\tilde{\gamma }\in \text {SL}(n+1,\mathbb {C})$ such that $\tilde{\gamma }$ is not diagonalizable with all its eigenvalue of norm one.

This classification coincides with the one of [6] for elements of $\text {PU}(1,n)$.

In our case, for elements $\gamma \in \text {PU}(r,s)$, we will say that $\gamma$ is loxodromic (parabolic, elliptic) if $\gamma$ seen as an element of $\text {PSL}(n+1,\mathbb {C})$ is loxodromic (parabolic, elliptic).

2.5 Chen-Greenberg Limit Set

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup acting on the complex hyperbolic space, $\mathbb {H}^{1,n}_{\mathbb {C}}$. See [6].

Definition 2.4

Let $\Gamma$ be a discrete subgroup of $\text {PU}(1,n)$ and $w\in \mathbb {H}^{1,n}_{\mathbb {C}}$. The limit set of $\Gamma$ in the sense of Chen-Greenberg, denoted $L(\Gamma )$, is

$$\begin{aligned} L(\Gamma )= \overline{\Gamma (w)} \cap \partial \mathbb {H}^{1,n}_{\mathbb {C}}. \end{aligned}$$

So, $L(\Gamma )$ is the $\Gamma$-invariant closed set contained in the boundary of the ball $\mathbb {H}^{1,n}_{\mathbb {C}}$, given by the accumulation points of orbits of points in $\mathbb {H}^{1,n}_{\mathbb {C}}$. The set $L(\Gamma )$ does not depend on w by Lemma 4.3.1 in [6].

Moreover, the action on $L(\Gamma )$ is minimal.

Lemma 2.1

(Lemma 4.3.3 in [6]) Let $\Gamma$ be a non-elementary subgroup of $\text {PU}(1,n)$. If $X\subset \partial \mathbb {H}^{1,n}_{\mathbb {C}}$ is a $\Gamma$-invariant closed set containing more than one point, then $L(\Gamma ) \subset X$, and every orbit in $L(\Gamma )$ is dense in $L(\Gamma )$.

The following proposition will be useful:

Proposition 2.1

(Theorem 12.2.3 in [15]) Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup such that $L(\Gamma )$ contains at least two points, then there exists one loxodromic element in $\Gamma$.

2.6 Pseudo-Projective Transformations

We consider the space of linear transformations from $\mathbb {C}^{n+1}$ to $\mathbb {C} ^{n+1}$, denote by $\textrm{M}(n+1,\mathbb {C})$. Let $N: \mathbb {C}^{n+1}\longrightarrow \mathbb {C}^{n+1}$ be a non-zero linear transformation and Ker(N) be its kernel. We denote by Ker([N]) the respective projectivization. Then, N induces a well defined map $[N]: \mathbb{C}\mathbb{P}^{n}- Ker([N]) \longrightarrow \mathbb{C}\mathbb{P}^{n}$ given by

$$\begin{aligned}{}[N]([v])=[N(v)]. \end{aligned}$$

We define the image of [N] as: $Im([N])=[N(\mathbb {C}^{n+1})-\{0\}]$. We define the space of pseudo-projective maps as follows: $QP(n+1,\mathbb {C})=\left( \textrm{M} (n+1,\mathbb {C}) - \{0\}\right) / \mathbb {C}^{*}$. These transformations were introduced in [5].

In what follows, we will say that the sequence $(\gamma _{m}) \subset \text {PSL}(n+1,\mathbb {C})$ converges to $\gamma \in QP(n+1,\mathbb {C})$ in the sense of pseudo-projective transformations if $\gamma _{m}\underset{m\rightarrow \infty }{\longrightarrow }\gamma$ uniformly on compact sets of $\mathbb{C}\mathbb{P}^{n} - Ker(\gamma )$. See Example 2 in Section 2.2.

The following proposition shows that we can find sequences in $QP(n+1,\mathbb {C})$ such that the convergence as a sequence of points in a projective space coincides with the convergence as a sequence of functions.

Proposition 2.2

(Proposition 3.1 in [5]) Let $(\gamma _{m})\subset \text {PSL}(n+1,\mathbb {C})$ be a sequence of distinct elements, then

• (1) there are a subsequence $(\tau _{m})\subset (\gamma _{m})$ and $\tau _{0}\in \textrm{M}(n+1,\mathbb {C})-\{0\}$ such that $\tau _{m}\underset{m\rightarrow \infty }{\longrightarrow }\ \tau _{0}$ as points in $QP(n+1,\mathbb {C})$;

• (2) if $(\tau _{m})$ is the sequence given by the previous part of this lemma, then $\tau _{m}\underset{m\rightarrow \infty }{\longrightarrow }\ \tau _{0}$, as functions, uniformly on compact sets of $\mathbb{C}\mathbb{P}^{n}- Ker(\tau _{0})$.

2.7 The Equicontinuity Region

Definition 2.5

Let $\Gamma$ be a group acting on a manifold X. The equicontinuity region of $\Gamma$, denoted $\mathrm {Eq(\Gamma )}$, is the set of points $z\in X$ for which there is an open neighbourhood U of z such that $\Gamma \mid _{U}$ is a normal family.

Recall that a collection of transformations is a normal family if and only if every sequence of distinct elements has a subsequence which converges uniformly on compact sets.

It is not hard to see that $\mathrm {Eq(\Gamma )}$ is an open $\Gamma$-invariant set. Let $\Gamma \subset \text {PSL}(n+1,\mathbb {C})$ be a discrete subgroup acting on $\mathbb{C}\mathbb{P}^{n}$, we define

$$\begin{aligned} Lim(\Gamma ):= \lbrace \gamma \in QP(n+1,\mathbb {C}): \exists (\gamma _{m}) \subset \Gamma \text{ such } \text{ that } \hspace{2mm} \gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \gamma \rbrace \end{aligned}$$

where $\gamma _{m}\underset{m\rightarrow \infty }{\longrightarrow }\ \gamma$ in the sense of pseudo-projective transformations and $(\gamma _{m})$ is a sequence of distinct elements.

Proposition 2.3

(Proposition 2.5 and Corollary 2.6 in [3]) Let $\Gamma \subset \text {PSL}(n+1,\mathbb {C})$ be a discrete subgroup, then

$$\begin{aligned} \text {Eq}(\Gamma )= \mathbb{C}\mathbb{P}^{n}- \overline{\displaystyle \bigcup _{\gamma \in Lim(\Gamma )}Ker(\gamma )}, \end{aligned}$$

and $\Gamma$ acts properly discontinuously on $\text {Eq}(\Gamma )$. Moreover, $\text {Eq}(\Gamma )\subset \Omega _{Kul}(\Gamma )$.

2.8 Kulkarni Limit Set

When one considers the action of a group on a topological space, there are no natural notions of limit set. Hence, we study the notion introduced by Kulkarni.

Definition 2.6

$See$ [13]. Let $\Gamma < \text {PSL}(n+1,\mathbb {C})$ be a subgroup. We define

• (1) The set $\mathcal {L}_{0} (\Gamma )$ as the closure of set of points with infinite isotropy.

• (2) The set $\mathcal {L}_{1}(\Gamma )$ as the closure of set of cluster points of $\Gamma z$ where z runs over all points in $\mathbb{C}\mathbb{P}^{n}- \mathcal {L}_{0}(\Gamma )$.

• (3) The set $\mathcal {L}_{2}(\Gamma )$ as the closure of the set of cluster points of $\Gamma K$ where K runs over all the compact subsets in $\mathbb{C}\mathbb{P}^{n} - (\mathcal {L}_{0}(\Gamma ) \cup \mathcal {L}_{1}(\Gamma ))$.

The Kulkarni limit set of $\Gamma$ is defined as follows:

$$\begin{aligned} \Lambda _{Kul}(\Gamma )=\mathcal {L}_{0}(\Gamma )\cup \mathcal {L}_{1}(\Gamma )\cup \mathcal {L}_{2}(\Gamma ); \end{aligned}$$

and the Kulkarni discontinuity region as follows:

$$\begin{aligned} \Omega _{Kul}(\Gamma )= \mathbb{C}\mathbb{P}^{n} - \Lambda _{Kul}(\Gamma ). \end{aligned}$$

The Kulkarni limit set has the following properties (see [4, 5, 13]).

Proposition 2.4

Let $\Gamma < \text {PSL}(n+1,\mathbb {C})$ be a discrete subgroup. Then, the following hold:

• (1) The sets $\Lambda _{Kul}(\Gamma ), \mathcal {L}_{0}(\Gamma ), \mathcal {L}_{1}(\Gamma ), \mathcal {L}_{2}(\Gamma )$ are $\Gamma$-invariant closed sets.

• (2) The group $\Gamma$ acts properly discontinuously on $\Omega _{Kul}(\Gamma )$.

• (3) The equicontinuity set of $\Gamma$ is contained in $\Omega _{Kul}(\Gamma )$.

• (4) If $\Gamma _{0} \subset \Gamma$ is a finite index subgroup, then $\Lambda _{Kul}(\Gamma _{0})=\Lambda _{Kul}(\Gamma )$.

The Kulkarni limit set for discrete subgroups of $\text {PU}(1,n)$ acting on $\mathbb{C}\mathbb{P}^{n}$ is described as follows:

Theorem 2.5

(Theorem 0.1 in [2]) Let $\Gamma \subset \text {PU}(1,n)$ be a non-elementary discrete subgroup, then the following hold:

• (1) The Kulkarni discontinuity region $\Omega _{Kul}(\Gamma )$ is the largest open set on which $\Gamma$ acts properly and discontinuously.

• (2) The set $\Omega _{Kul}(\Gamma )$ coincides with the equicontinuity set $\text {Eq}(\Gamma )$.

• (3) The Kulkarni limit set of $\Gamma$ can be described as the union of hyperplanes tangent to $\partial \mathbb {H}^{1,n}_{\mathbb {C}}$ at points in the Chen-Greenberg limit set of $\Gamma$, $L(\Gamma )$, that is,

  $$\Lambda _{Kul}(\Gamma )=\bigcup _{p\in L(\Gamma )}p^{\perp }.$$

2.9 Cartan Decomposition

The following version of the Cartan decomposition $G=KAK$ for the group $\text {PU}(1,n)$ can be found in [14] (see Theorem 3.4 and problem 3.7, chapter 2).

Theorem 2.6

(Cartan decomposition) For every $\gamma \in \text {PU}(1,n)$ there are elements $g_{}, h_{} \in K=\text {PU}(1,n)\cap \text {PU}(1+n)$ and a unique $\mu (\gamma )\in \text {PU}(1,n)$, such that $\gamma =g_{}\mu (\gamma )h_{}$ and $\mu (\gamma )$ has a lift in $\text {SL}(n+1,\mathbb {C})$ given by

$$\begin{aligned} \begin{pmatrix} e^{\lambda (\gamma )}&{} &{} &{} &{} \\ &{} 1 &{} &{} &{}\\ &{} &{} \ddots &{} &{}\\ &{} &{} &{} 1 &{} \\ &{} &{} &{} &{} e^{-\lambda (\gamma )} \end{pmatrix} \end{aligned}$$

where $\lambda (\gamma ) \ge 0$.

Recall that a sequence $(\gamma _{m})\subset \text {PSL}(n+1,\mathbb {C})$ is divergent if it leaves every compact set, that is, for each compact set $C\subset$ GL$(n+1,\mathbb {C}$) exists $N\in \mathbb {N}$ such that for each $m\ge N$, $\gamma _{m} \notin C$.

We follow the Definition 3.1 in [13] or Definition 3.3 in [1].

Definition 2.7

Let $(\gamma _{m})\subset \text {PU}(1,n)$ be a divergent sequence of distinct elements. We will say that $(\gamma _{m})$ tends simply to infinity if the sequence $\lambda (\gamma _{m})$ in the Cartan decomposition converges to infinity.

Given a discrete group $\Gamma \subset \text {PU}(1,n)$ and a divergent sequence $(\gamma _{m})\subset \Gamma$ of distinct elements, there is a subsequence $(\tau _{m})\subset (\gamma _{m})$ tending simply to infinity.

3 Geometry of Exterior Powers of $\mathbb {C}^{n+1}$

Let $\textit{V}=\lbrace v_{i}: i\in \lbrace 1,...,n+1 \rbrace \rbrace$ be an ordered basis of $\mathbb {C}^{n+1}$, then

$$\begin{aligned} \lbrace v_{i_{1}}\wedge ...\wedge v_{i_{k+1}} \mid 1\le i_{1}<...< i_{k+1}\le n+1 \rbrace \end{aligned}$$

is a lexicographically ordered basis $\left\{ v_{I} \right\}$ of $\wedge ^{k+1}\mathbb {C}^{n+1}$, where $v_{I}=v_{i_{1}}\wedge ...\wedge v_{i_{k+1}}$ with index $I=(i_{1},...,i_{k+1})$. Define

$$\begin{aligned} I(k+1,n+1)=\{I=(i_{1},...,i_{k+1})\in \mathbb {Z}^{k+1}: 1\le i_{1}<...<i_{k+1}\le n+1\}. \end{aligned}$$

3.1 The Grassmannians

Let $0 \le k \le n$, we define $\text {Gr}_{k+1}(\mathbb {C}^{n+1})$ the Grassmannian as the space of all $(k+1)$-dimensional linear subspaces of $\mathbb {C}^{n+1}$. Its projective version, $\text {Gr}_{k}(\mathbb{C}\mathbb{P}^{n})$, is the space of all k-dimensional projective subspaces of $\mathbb{C}\mathbb{P}^{n}$ endowed with the Hausdorff topology. So, $\text {Gr}_{k}(\mathbb{C}\mathbb{P}^{n})$ is a complex, compact, and connected manifold of dimension $(k+1)(n-k)$.

A way to realize the Grassmannian as a subvariety of the projective space of the $(k + 1)$-exterior power of $\mathbb {C}^{n+1}$, in symbols $\mathbb {P}( \wedge ^{k+1} \mathbb {C}^{n+1})$, is done by the Plücker embedding, which is given by the following:

$$\begin{aligned} \begin{array}{rccl} P_{k,n}:&{} \text {Gr}_{k}(\mathbb{C}\mathbb{P}^{n}) &{}\longrightarrow &{}\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})\\ &{}V &{}\mapsto &{} [v_{1}\wedge v_{2}\wedge ... \wedge v_{k+1}] \end{array} \end{aligned}$$

where $V=[Span \lbrace v_{1}, v_{2},...,v_{k+1} \rbrace ]$. Let $Q_{k,n}\subset \mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$ be the image under Plücker embedding of the Grassmannian $\text {Gr}_{k}(\mathbb{C}\mathbb{P}^{n})$.

Now, we see some linear subvarieties of the vector space $\wedge ^{k+1}\mathbb {C}^{n+1}$ generated by certain subsets of $\text {Gr}_{k+1}(\mathbb {C}^{n+1})$; for that, let us define the following sets:

$$\begin{aligned} \pi _{\alpha }^{k+1}(W)=\lbrace \ell \in \text {Gr}_{k+1}(\mathbb {C}^{n+1}): W \subset \ell \rbrace , \;\textrm{with}\; W\in \text {Gr}_{s+1}(\mathbb {C}^{n+1}) \;\textrm{for}\; s\le k, \end{aligned}$$

that is, the set of all $(k+1)$-subspaces containing a $(s+1)$-subspace fixed, $W\subset \mathbb {C}^{n+1}$;

$$\begin{aligned} \pi _{\beta }^{k+1}(U)=\lbrace \ell \in \text {Gr}_{k+1}(\mathbb {C}^{n+1}): \ell \subset U \rbrace ,\text { with }U \in \text {Gr}_{t+1}(\mathbb {C}^{n+1})\text { and }t\ge k, \end{aligned}$$

that is, the set of all $(k+1)$-subspaces that are contained in a $(t+1)$-subspace fixed, $U\subset \mathbb {C}^{n+1}$. These sets generate linear subvarieties in $\wedge ^{k+1}\mathbb {C}^{n+1}$ of dimension $\left( {\begin{array}{c}n-s\\ k-s\end{array}}\right)$ and $\left( {\begin{array}{c}t+1\\ k+1\end{array}}\right)$, respectively, by the Plücker embedding.

Example 1

We consider $\text {Gr}_{2}(\mathbb {C}^{4})$: Given $\{a,b,c,d\}$ a basis of $\mathbb {C}^{4}$ and the hyperplane $V=Span\{a,b,c\}$, so

$$\begin{aligned} \begin{aligned} \pi _{\alpha }^{2}(a)= & {} \lbrace \ell \in \text {Gr}_{2}(\mathbb {C}^{4}): a \in \ell \rbrace ,\\ \pi _{\beta }^{2}(V)= & {} \lbrace \ell \in \text {Gr}_{2}(\mathbb {C}^{4}): \ell \subset V \rbrace . \end{aligned} \end{aligned}$$

The above sets generate the following linear subvarieties of the vector space $\wedge ^{2}\mathbb {C}^{4}$ by the Plücker embedding:

$$\begin{aligned} \begin{aligned} \left\langle \pi _{\alpha }^{2}(a) \right\rangle= & {} Span \{ a\wedge b, a\wedge c, a\wedge d \}, \\ \left\langle \pi _{\beta }^{2}(V)\right\rangle= & {} Span\{ a\wedge b, a\wedge c, b\wedge c \}. \end{aligned} \end{aligned}$$

3.2 The Induced Action

We consider the following homomorphism:

$$\begin{aligned} \begin{array}{rccl} \phi _{k}:&{} \text {PSL}(n+1,\mathbb {C})&{}\longrightarrow &{}\mathbb {P} \left( \text{ Aut }(\wedge ^{k+1}\mathbb {C}^{n+1})\right) \\ &{}[\gamma ]&{}\mapsto &{}[ \wedge ^{k+1}\gamma ]. \end{array} \end{aligned}$$

The morphism $\phi _{k}$ is an injective homomorphism of groups; see [9].

Moreover, this morphism maps discrete subgroups into discrete subgroups:

Theorem 3.1

(Theorem 2.1 and Theorem 2.3 in [16]) Let $\Gamma \subset \text {PSL}(n+1,\mathbb {C})$ be a discrete subgroup, $0< k +1 < n+1$, we have the following embedding that carries discrete groups into discrete groups:

$$\begin{aligned} \begin{array}{rccl} \phi _{k}:&{} \text {PU}(1,n)&{}\longrightarrow &{} \text {PU}(R,S)\\ &{} \Gamma &{}\mapsto &{} \wedge ^{k+1}\Gamma . \end{array} \end{aligned}$$

where $R=\left( {\begin{array}{c}n\\ k\end{array}}\right)$ and $S=\left( {\begin{array}{c}n\\ k+1\end{array}}\right) .$

From Theorem 3.3, Corollary 3.7, and Corollary 3.8 in [16], we have the following result:

Theorem 3.2

Let $(\gamma _{m}) \subset \text {U}(1,n)$ be a sequence tending simply to infinity. Then, for $0 \le k+1<n+1$, there are two pseudo-projective transformations $\rho , \delta \in QP\left( \left( {\begin{array}{c}n+1\\ k+1\end{array}}\right) , \mathbb {C} \right)$ such that $[\wedge ^{k+1}\gamma _{m}]\underset{m\rightarrow \infty }{\longrightarrow }\ [\rho ]$ and $\hspace{1mm}[\wedge ^{k+1}\gamma _{m}^{-1}]\underset{m\rightarrow \infty }{\longrightarrow }[\delta ]$ in the sense of pseudo-projective transformations. Given $\gamma _{m}= g^{}_{m}\mu (\gamma _{m})h^{}_{m}$ by the Cartan decomposition theorem, then

$$\begin{aligned} \begin{aligned} Im(\rho )&=\wedge ^{k+1}g\left( Span\{ e_{I} \} : I\in A\right) ,\\ Im(\delta )&= \wedge ^{k+1}h^{-1}\left( Span\{ e_{I}\}: I\in B \right) ,\\ Ker(\rho )&= \wedge ^{k+1}h^{-1}\left( Span\{ e_{I}\}: I \notin A \right) ,\\ Ker(\delta )&=\wedge ^{k+1}g\left( Span\{ e_{I}\}: I\notin B \right) , \end{aligned} \end{aligned}$$

(3.1)

where $g,h \in U(1,n)\cap U(1+n)$ such that $g_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ g, \hspace{2mm} \hspace{2mm} h_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ h$, and

$$\begin{aligned} \begin{aligned} A= & {} \{I=(1,i_{2},...,i_{k+1}) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne n+1\},\\ B= & {} \{I=(i_{1},...,i_{k},n+1) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne 1\}. \end{aligned} \end{aligned}$$

Given $p, q \in \mathbb {C}^{n+1}$ such that $[\gamma _{m}]\underset{m\rightarrow \infty }{\longrightarrow }\ [\tau ]$ and $[\gamma _{m}^{-1}]\underset{m\rightarrow \infty }{\longrightarrow }\ [\theta ]$ in the sense of pseudo-projective transformations, where $Im(\tau )=\{p\}$ and $Im(\theta )=\{q\}$, then

$$\begin{aligned} \begin{aligned} Im(\rho )&= Span\{ \pi ^{k+1}_{\alpha }(p) \cap \pi ^{k+1}_{\beta }(p^{\perp })\},\\ Im(\delta )&= Span\{ \pi ^{k+1}_{\alpha }(q) \cap \pi ^{k+1}_{\beta }(q^{\perp })\},\\ Ker(\rho )&=Span\{\pi ^{k+1}_{\alpha }(q), \pi ^{k+1}_{\beta }(q^{\perp }) \},\\ Ker(\delta )&=Span\{ \pi ^{k+1}_{\alpha }(p), \pi ^{k+1}_{\beta }(p^{\perp }) \}. \end{aligned} \end{aligned}$$

(3.2)

Example 2

Let us consider the element $\gamma \in \text {PU}(1,3)$ induced by the matrix

$$\begin{aligned} \tilde{ \gamma }= \begin{pmatrix} 2&{} 0 &{} 0 &{} 0 \\ 0&{} 1 &{} 0 &{} 0 \\ 0&{} 0 &{} 1 &{} 0 \\ 0&{} 0 &{} 0 &{} 2^{-1} \end{pmatrix} \end{aligned}$$

and $\Gamma =\left\langle \gamma \right\rangle$. Then, $\gamma ^{m}\underset{m\rightarrow \infty }{\longrightarrow }\ [\tau ]$ and $\gamma ^{-m}\underset{m\rightarrow \infty }{\longrightarrow }\ [\theta ]$ in the sense of pseudo-projective transformations, where

$$\begin{aligned} \tau =\begin{pmatrix} 1&{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 \end{pmatrix} \textrm{and}\ \theta =\begin{pmatrix} 0&{} 0 &{} 0 &{} 0 \\ 0&{}0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 1 \end{pmatrix}, \end{aligned}$$

with

$$\begin{aligned} Im(\tau )= & {} \{ e_{1}\},\\ Im(\theta )= & {} \{ e_{4}\},\\ Ker(\tau )= & {} Span\{ e_{2}, e_{3}, e_{4}\}=e_{4}^{\perp } ,\\ Ker(\theta )= & {} Span\{ e_{1}, e_{2}, e_{3}\}=e_{1}^{\perp }. \end{aligned}$$

Where $\{e_{1}, e_{2}, e_{3}, e_{4}\}$ is the standard basis of $\mathbb {C}^{4}$, it induces the ordered basis $\{ e_{1,2},e_{1,3},e_{1,4},e_{2,3},e_{2,4},e_{3,4}\}$ of $\wedge ^{2}\mathbb {C}^{4}$.

Now,

$$\begin{aligned} \wedge ^{2}\tilde{\gamma }= \begin{pmatrix} 2&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 2 &{} 0 &{} 0 &{} 0 &{} 0\\ 0&{} 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 0&{} 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0&{} 0 &{} 0 &{} 0 &{} 2^{-1} &{} 0\\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 2^{-1} \end{pmatrix}\in \text {U}(3,3). \end{aligned}$$

So,

$$\begin{aligned} \wedge ^{2}\gamma ^{m}\underset{m\rightarrow \infty }{\longrightarrow }\ [\rho ]=\left[ \begin{pmatrix} 1&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{pmatrix} \right] \end{aligned}$$

and

$$\begin{aligned} \wedge ^{2}\gamma ^{-m}\underset{m\rightarrow \infty }{\longrightarrow }\ [\delta ]=\left[ \begin{pmatrix} 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} \right] \end{aligned}$$

in the sense of pseudo-projective transformations, where

$$\begin{aligned} \begin{aligned} Im(\rho )&=Span\{e_{1,2}, e_{1,3}\}\\&=Span\{{e}_{1}\wedge {e}_{2} , {e}_{1}\wedge {e}_{3}\}\\&= Span\{ \pi ^{2}_{\alpha }(e_{1}) \cap \pi ^{2}_{\beta }(e_{1}^{\perp })\},\\ Im(\delta )&=Span\{e_{2,4}, e_{3,4}\}\\&=Span\{{e}_{2}\wedge {e}_{4} , {e}_{3}\wedge {e}_{4}\}\\&= Span\{ \pi ^{2}_{\alpha }(e_{4}) \cap \pi ^{2}_{\beta }(e_{4}^{\perp })\},\\ Ker(\rho )&=Span\{e_{1,4}, e_{2,3},e_{2,4}, e_{3,4}\}\\&=Span\{{e}_{1}\wedge {e}_{4} , {e}_{2}\wedge {e}_{3},{e}_{2}\wedge {e}_{4} , {e}_{3}\wedge {e}_{4}\}\\&= Span\{ \pi ^{2}_{\alpha }(e_{4}) \cup \pi ^{2}_{\beta }(e_{4}^{\perp })\},\\ Ker(\delta )&=Span\{e_{1,2}, e_{1,3},e_{1,4}, e_{2,3}\}\\&=Span\{{e}_{1}\wedge {e}_{2} , {e}_{1}\wedge {e}_{3},{e}_{1}\wedge {e}_{4} , {e}_{2}\wedge {e}_{3}\}\\&= Span\{ \pi ^{2}_{\alpha }(e_{1}) \cup \pi ^{2}_{\beta }(e_{1}^{\perp })\}. \end{aligned} \end{aligned}$$

3.3 Induced Hermitian Product

Given a Hermitian product in $\mathbb {C}^{n+1}$ defined by the matrix $H_{1,n}$, it induces scalar product on the $(k+1)$-th exterior power of $\mathbb {C}^{n+1}$ by setting the Hermitian matrix $H_{R,S}=\wedge ^{k+1}H_{1,n}$.

Since any $w\in \wedge ^{k+1}\mathbb {C}^{n+1}$ is written as a linear combination over $\mathbb {C}$,

$$\begin{aligned} w=\sum _{I\in I(k+1,n+1)}w^{I}v_{I},\hspace{2mm} w^{I}\in \mathbb {C}. \end{aligned}$$

So, we just need to analyze the Hermitian product for $(k+1)$-vectors of the form $v_{1}\wedge ...\wedge v_{k+1}$, with $\{v_{1},...,v_{k+1}\}\subset \mathbb {C}^{n+1}$ linearly independent vectors.

We need the following result of linear algebra:

Proposition 3.3

(See [7]) Given vectors $u_{1},...,u_{m}, v_{1},...,v_{m} \in \mathbb {C}^{n}$ we have:

$$\begin{aligned} \langle v_{1}\wedge ...\wedge v_{m}, u_{1}\wedge ...\wedge u_{m}\rangle = \textrm{det} \left( \begin{array}{rccl} \langle v_{1},u_{1}\rangle &{}\langle v_{2},u_{1}\rangle &{}...&{} \langle v_{m}, u_{1}\rangle \\ \langle v_{1},u_{2}\rangle &{}\langle v_{2},u_{2}\rangle &{}...&{} \langle v_{m}, u_{2}\rangle \\ \vdots &{} \vdots &{}\ddots &{} \vdots \\ \langle v_{1},u_{m}\rangle &{}\langle v_{2},u_{m}\rangle &{}...&{} \langle v_{m}, u_{m}\rangle \end{array} \right) \end{aligned}$$

where $\langle .,.\rangle$ is the standard scalar product.

Now, we describe the induced product on $\wedge ^{k+1}\mathbb {C}^{n+1}$ given by the Hermitian matrix $H_{R,S}$, defined by

$$\begin{aligned} \langle z, w \rangle _{R,S}=z H_{R,S}w^{*} \end{aligned}$$

for all vectors $z,w \in \wedge ^{k+1}\mathbb {C}^{n+1}$.

Proposition 3.4

Given vectors $z_{1},...,z_{k+1}, w_{1},...,w_{k+1} \in \mathbb {C}^{n+1}$ we have the following:

$$\begin{aligned} \langle z_{1}\wedge ...\wedge z_{k+1}, w_{1}\wedge ...\wedge w_{k+1}\rangle _{R,S} = det \left( \begin{array}{rccl} \langle z_{1},w_{1}\rangle _{1,n} &{}...&{} \langle z_{k+1}, w_{1}\rangle _{1,n}\\ \vdots &{}\ddots &{} \vdots \\ \langle z_{1},w_{k+1}\rangle _{1,n} &{}...&{} \langle z_{k+1}, w_{k+1}\rangle _{1,n} \end{array} \right) \end{aligned}$$

where $\langle .,.\rangle _{1,n}$ is the Hermitian product defined by $H_{1,n}$ on $\mathbb {C}^{n+1}$.

Proof

Let $z=z_{1}\wedge ...\wedge z_{k+1}, w=w_{1}\wedge ...\wedge w_{k+1} \in \wedge ^{k+1}\mathbb {C}^{n+1}$ be vectors,

$$\begin{aligned} \begin{aligned} \langle z,w \rangle _{R,S}&= z H_{R,S} w^{*}\\&= z H_{R,S} (w_{1}\wedge ...\wedge w_{k+1})^{*}\\&= z H_{R,S} ( \overline{w_{1}}\wedge ...\wedge \overline{w_{k+1}})^{t}\\&= z (H_{1,n} (w_{1}^{*})\wedge ...\wedge H_{1,n} (w_{k+1}^{*}))^{t}\\&=\left\langle z, \overline{H_{1,n} (w_{1}^{*})\wedge ...\wedge H_{1,n} (w_{k+1}^{*})}\right\rangle \\&=\left\langle z, \overline{H_{1,n} (w_{1}^{*})}\wedge ...\wedge \overline{H_{1,n} (w_{k+1}^{*})}\right\rangle . \end{aligned} \end{aligned}$$

By Proposition 3.3, we get the following:

$$\begin{aligned} \left\langle z, \overline{H_{1,n} (w_{1}^{*})}\wedge ...\wedge \overline{H_{1,n} (w_{k+1}^{*})}\right\rangle =\text {det} \left( \begin{array}{rccl} \langle z_{1},\overline{H_{1,n} (w_{1}^{*})} \rangle &{}...&{} \langle z_{k+1}, \overline{H_{1,n} (w_{1}^{*})}\rangle \\ \vdots &{}\ddots &{} \vdots \\ \langle z_{1},\overline{H_{1,n} (w_{k+1}^{*})}\rangle &{}...&{} \langle z_{k+1}, \overline{H_{1,n} (w_{k+1}^{*})}\rangle \end{array} \right) \end{aligned}$$

where $\langle z_{i}, \overline{H_{1,n} (w_{j}^{*})}\rangle = z_{i} H_{1,n} w_{j}^{*}=\langle z_{i},w_{j} \rangle _{1,n}$. Then

$$\begin{aligned} \langle z,w \rangle _{R,S} = \textrm{det} \left( \begin{array}{rccl} \langle z_{1},w_{1}\rangle _{1,n} &{}...&{} \langle z_{k+1}, w_{1}\rangle _{1,n}\\ \vdots &{}\ddots &{} \vdots \\ \langle z_{1},w_{k+1}\rangle _{1,n} &{}...&{} \langle z_{k+1}, w_{k+1}\rangle _{1,n} \end{array} \right) \end{aligned}$$

3.4 Induced Linear Subvarieties

Let [p] be a point in $\partial \mathbb {H}^{1,n}_{\mathbb {C}}$, $0\le k < n$. We consider

$$\begin{aligned} \begin{aligned} \pi _{\alpha }^{k+1}(p)&=\lbrace \ell \in \text {Gr}_{k+1}(\mathbb {C}^{n+1}): p \in \ell \rbrace ,\\ \pi _{\beta }^{k+1}(p^{\perp })&=\lbrace \ell \in \text {Gr}_{k+1}(\mathbb {C}^{n+1}): \ell \subset p^{\perp } \rbrace . \end{aligned} \end{aligned}$$

These sets generate linear subvarieties of the vector space $\wedge ^{k+1}\mathbb {C}^{n+1}$ via the Plücker embedding. Now, we define

$$\begin{aligned} \begin{aligned} H^{k}_{p}&:=\pi ^{k+1}_{\alpha }(p)\cap \pi ^{k+1}_{\beta }(p^{\perp })\\&=\{ \ell \in \text {Gr}_{k+1}(\mathbb {C}^{n+1}): p\in \ell \subset p^{\perp }\}. \end{aligned} \end{aligned}$$

We will geometrically describe the linear subvarieties that generate the following sets $H^{k}_{p}, \pi ^{k+1}_{\alpha }(p)$, and $\pi ^{k+1}_{\beta }(p^{\perp })$ in the vector space $\wedge ^{k+1}\mathbb {C}^{n+1}$ via the Plücker embedding. These sets are key to the next section.

Given p and q be two different vectors in $V_{0}^{1,n}$, let $\{a_{2},...,a_{n}\}$ be an orthogonal basis of $p^{\perp } \cap q^{\perp }$, that is, $p^{\perp }\cap q^{\perp }=Span\{a_{2},...,a_{n}\}$. Then, $U=\{p,a_{2},...,a_{n},q\}$ is an ordered basis of $\mathbb {C}^{n+1}$ such that

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{cl} \langle p,a_{i}\rangle _{1,n}=0 =\langle q,a_{i}\rangle _{1,n} &{}\text{ for } i=2,...,n\\ \langle p, p\rangle _{1,n}=0=\langle q,q \rangle _{1,n}\\ \langle a_{i},a_{j}\rangle _{1,n} \left\{ \begin{array}{cl} >0 &{}\text{ if } i=j\\ =0 &{}\text{ if } i\ne j. \end{array}\right. \end{array}\right. \end{aligned} \end{aligned}$$

(3.3)

This basis induces a basis in $\wedge ^{k+1}\mathbb {C}^{n+1}$ of the following linear subvarieties:

$$\begin{aligned} \begin{aligned} \left\langle H^{k}_{p} \right\rangle&=Span \{ u_{I}: I=(1,i_{1},...,i_{k}) \in I(k+1,n+1)\hspace{2mm} i_{k}\ne n+1\},\\ \left\langle \pi _{\alpha }^{k}(p)\right\rangle&=Span\{ u_{I}: I=(1,i_{1},...,i_{k}) \in I(k+1,n+1)\},\\ \left\langle \pi _{\beta }^{k}(p^{\perp })\right\rangle&=Span\{ u_{I}: I=(i_{1},...,i_{k+1}) \in I(k+1,n+1)\hspace{2mm} i_{k+1}\ne n+1\},\\ \left\langle \pi _{\alpha }^{k}(p), \pi _{\beta }^{k}(p^{\perp }) \right\rangle&=Span \{ u_{I}:I=(i_{1},...,i_{k+1})\hspace{2mm} i_{j}\ne n+1\hspace{2mm} or\hspace{2mm} I=(1,i_{1},...,i_{k})\}. \end{aligned} \end{aligned}$$

Note that the matrix used to compute the various products has many vanishing coefficients due to the choice of this basis.

Consider $H^{k}_{p}\subset \wedge ^{k+1}\mathbb {C}^{n+1}$ via the Plücker embedding, so

$$\begin{aligned} \langle H^{k}_{p}\rangle =Span\{u_{I}: u_{I}=p\wedge a_{i_{1}}\wedge ...\wedge a_{i_{k}}\}. \end{aligned}$$

Therefore, $dim\langle H^{k}_{p}\rangle = \left( {\begin{array}{c}n-1\\ k\end{array}}\right)$.

Proposition 3.5

The vector space $\langle H^{k}_{p}\rangle$ is contained in $V^{R,S}_{0}$.

Proof

Let $u_{I}=p\wedge a_{i_{1}}\wedge ...\wedge a_{i_{k}}, u_{J}=p\wedge a_{j_{1}}\wedge ...\wedge a_{j_{k}}$ be two different points in $H^{k}_{p}$, where $I=(1,i_{1},...,i_{k})$ and $J=(1,j_{1},...,j_{k})$, $I\ne J$, $I,J\in I(k+1,n+1)$. By Proposition 3.4,

$$\begin{aligned} \langle u_{I},u_{I} \rangle _{R,S}=0=\langle u_{I},u_{J} \rangle _{R,S}. \end{aligned}$$

From the above and the bilinearity of $\langle .,.\rangle _{R,S}$, we conclude that $\langle H^{k}_{p}\rangle \subset V^{R,S}_{0}$ as a vector space.

Consider $\pi _{\alpha }^{k+1}(p)\subset \wedge ^{k+1}\mathbb {C}^{n+1}$ via the Plücker embedding, so

$$\begin{aligned} \langle \pi ^{k+1}_{\alpha }(p)\rangle =Span\{u_{I} : u_{I}=p\wedge a_{i_{1}}\wedge ...\wedge a_{i_{k}} \hspace{2mm} \textrm{or} \hspace{2mm} u_{I}=p \wedge a_{i_{1}}\wedge ...\wedge a_{i_{k-1}}\wedge q\}. \end{aligned}$$

Therefore, $dim\langle \pi ^{k+1}_{\alpha }(p)\rangle = \left( {\begin{array}{c}n\\ k\end{array}}\right)$.

Proposition 3.6

The vector space $\langle \pi ^{k+1}_{\alpha }(p)\rangle$ is contained in $\langle H^{k}_{p}\rangle ^{\perp }$. Moreover, $\langle \pi ^{k+1}_{\alpha }(p)\rangle \subset V^{R,S}_{0}\cup V^{R,S}_{-}$.

Proof

From the last proposition $\langle H^{k}_{p}\rangle \subset \langle \pi ^{k+1}_{\alpha }(p)\rangle$ is such that $\langle H^{k}_{p}\rangle \subset V^{R,S}_{0}$ as a vector space. Now, consider $u_{J}=p\wedge a_{j_{1}}\wedge ...\wedge a_{j_{k-1}}\wedge q$ for $J=(1,j_{1},...,j_{k-1},n+1)$, so, from Proposition 3.4

$$\begin{aligned} \begin{aligned} \langle u_{J},u_{J} \rangle _{R,S}&=det\left( \begin{array}{rccl} \langle p,p\rangle _{1,n} &{}\langle a_{j_{1}},p\rangle _{1,n}&{} \cdots &{} \langle q,p\rangle _{1,n}\\ \langle p,a_{j_{1}}\rangle _{1,n} &{}\langle a_{j_{1}},a_{j_{1}}\rangle _{1,n}&{}\cdots &{} \langle q,a_{j_{1}}\rangle _{1,n}\\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ \langle p,q\rangle _{1,n}&{} \langle a_{j_{1}},q\rangle _{1,n}&{} \cdots &{} \langle q,q\rangle _{1,n} \end{array} \right) \\&=(-1)^{k+2}\langle q,p\rangle _{1,n}(-1)^{k+1}\langle p,q\rangle _{1,n}\langle a_{j_{1}},a_{j_{1}}\rangle _{1,n} \cdot \cdot \cdot \langle a_{j_{k-1}},a_{j_{k-1}}\rangle _{1,n}\\&=(-1)|\langle q,p\rangle _{1,n}|^{2}\langle a_{j_{1}},a_{j_{1}}\rangle _{1,n} \cdot \cdot \cdot \langle a_{j_{k-1}},a_{j_{k-1}}\rangle _{1,n} <0. \end{aligned} \end{aligned}$$

Therefore, for every $u_{I} \in \langle \pi ^{k+1}_{\alpha }(p)\rangle$, we have

$$\begin{aligned} u_{I}\in \left\{ \begin{array}{lcc} V^{R,S}_{0} &{} if &{} I=(1,i_{1},...,i_{k}) \hspace{2mm}\text{ with } \hspace{2mm} i_{j}\ne n+1 \\ \\ V^{R,S}_{-} &{} if &{} I=(1,i_{1},...,i_{k-1},n+1). \end{array} \right. \end{aligned}$$

Given $u_{J_{1}}=p\wedge a_{j_{1}}\wedge ...\wedge a_{j_{k-1}}\wedge q, u_{J_{2}}=p\wedge a_{i_{1}}\wedge ...\wedge a_{i_{k-1}}\wedge q \in \pi ^{k+1}_{\alpha }(p) -H^{k}_{p}$, with $J_{1}\ne J_{2}$; so, there is at least one $a_{j_{*}}$ such that $a_{j_{*}}\ne a_{i_{l}}$ for all $l\in \{i_{1},...,i_{k-1}\}$. Then, $\langle u_{J_{1}},u_{J_{2}} \rangle _{R,S}=0$ by equation (3.3) and Proposition 3.4.

On the other hand, for $u_{I}=p\wedge a_{i_{1}}\wedge ...\wedge a_{i_{k}}\in H^{k}_{p}\subset \pi ^{k+1}_{\alpha }(p)$ and $u_{J}=p\wedge a_{j_{1}}\wedge ...\wedge a_{j_{k-1}}\wedge q \in \pi ^{k+1}_{\alpha }(p) - H^{k}_{p}$, we get $\langle u_{I},u_{J} \rangle _{R,S}=0$ by Proposition 3.4.

From the above, Proposition 3.5 and the bilinearity of $\langle .,.\rangle _{R,S}$, we conclude that

$$\{u_{I}: I=(1,i_{1},...,i_{k}) \in I(k+1,n+1)\}$$

form an orthogonal basis of $\langle \pi ^{k+1}_{\alpha }(p) \rangle$, and $\langle \pi ^{k+1}_{\alpha }(p) \rangle \subset \langle H^{k}_{p}\rangle ^{\perp }$.

Let $x\in \langle \pi ^{k+1}_{\alpha }(p) \rangle$ be a point, then there are $\alpha _{i} \in \mathbb {C}$ such that

$$x=\alpha _{1} u_{I_{1}}+ \cdot \cdot \cdot + \alpha _{M} u_{I_{M}},$$

where $M=\left( {\begin{array}{c}n\\ k\end{array}}\right)$. So,

$$\begin{aligned} \langle x,x \rangle _{R,S}= |\alpha _{1}|^{2}\langle u_{I_{1}},u_{I_{1}}\rangle _{R,S}+\cdots + |\alpha _{M}|^{2}\langle u_{I_{M}},u_{I_{M}}\rangle _{R,S}\le 0. \end{aligned}$$

Moreover,

$$\begin{aligned} \langle x, x\rangle _{R,S}=0 \Leftrightarrow x\in \langle H^{k}_{p}\rangle . \end{aligned}$$

Hence, $\langle \pi ^{k+1}_{\alpha }(p) \rangle \subset V^{R,S}_{0}\cup V^{R,S}_{-}$ as a vector space.

Consider $\pi ^{k+1}_{\beta }(p^{\perp })\subset \wedge ^{k+1}\mathbb {C}^{n+1}$ via the Plücker embedding, so

$$\begin{aligned} \langle \pi ^{k+1}_{\beta }(p^{\perp })\rangle =Span\{u_{I} : u_{I}=p\wedge a_{i_{1}}\wedge ...\wedge a_{i_{k}} \hspace{2mm} \textrm{or} \hspace{2mm} u_{I}=a_{i_{1}}\wedge ...\wedge a_{i_{k+1}}\}. \end{aligned}$$

Therefore, $dim\langle \pi ^{k+1}_{\beta }(p^{\perp })\rangle = \left( {\begin{array}{c}n\\ k+1\end{array}}\right)$.

Proposition 3.7

The vector space $\langle \pi ^{k+1}_{\beta }(p^{\perp })\rangle$ is contained in $\langle H^{k}_{p}\rangle ^{\perp }$. Moreover, $\langle \pi ^{k+1}_{\beta }(p^{\perp })\rangle \subset V^{R,S}_{0}\cup V^{R,S}_{+}$.

Proof

From Proposition 3.5$\langle H^{k}_{p}\rangle \subset \langle \pi ^{k+1}_{\beta }(p^{\perp })\rangle$ is such that $\langle H^{k}_{p}\rangle \subset V^{R,S}_{0}$ as a vector space. Now, consider $u_{J}=a_{j_{1}}\wedge ...\wedge a_{j_{k+1}}$ with $J=(j_{1},...,j_{k+1})$ for $j_{l}\ne 1,n+1$. From Proposition 3.4

$$\begin{aligned} \begin{aligned} \langle u_{J},u_{J} \rangle _{R,S}&=det\left( \begin{array}{rccl} \langle a_{j_{1}},a_{j_{1}}\rangle _{1,n} &{}\langle a_{j_{2}},a_{j_{1}}\rangle _{1,n}&{} \cdots &{} \langle a_{j_{k+1}},a_{j_{1}}\rangle _{1,n}\\ \langle a_{j_{1}},a_{j_{2}} \rangle _{1,n} &{}\langle a_{j_{2}},a_{j_{2}}\rangle _{1,n}&{}\cdots &{} \langle a_{j_{k+1}},a_{j_{2}}\rangle _{1,n}\\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ \langle a_{j_{1}},a_{j_{k+1}}\rangle _{1,n}&{} \langle a_{j_{2}},a_{j_{k+1}}\rangle _{1,n}&{} \cdots &{} \langle a_{j_{k+1}},a_{j_{k+1}}\rangle _{1,n} \end{array} \right) \\&=\langle a_{j_{1}},a_{j_{1}}\rangle _{1,n} \cdot \cdot \cdot \langle a_{j_{k+1}},a_{j_{k+1}}\rangle _{1,n} > 0. \end{aligned} \end{aligned}$$

So, all $u_{I}\in \langle \pi ^{k+1}_{\beta }(p^{\perp })\rangle$ is such that

$$\begin{aligned} u_{I}\in \left\{ \begin{array}{lcc} V^{R,S}_{0} &{} if &{} I=(1,i_{1},...,i_{k}) \hspace{2mm}\text{ with } \hspace{2mm} i_{l}\ne n+1 \\ \\ V^{R,S}_{+} &{} if &{} I=(i_{1},...,i_{k+1}) \hspace{2mm}\text{ with } \hspace{2mm} i_{l}\ne 1,n+1. \end{array} \right. \end{aligned}$$

Given $u_{J_{1}}=a_{i_{1}}\wedge ...\wedge a_{i_{k+1}}, u_{J_{2}}=a_{j_{1}}\wedge ...\wedge a_{j_{k+1}}\in \pi ^{k+1}_{\beta }(p^{\perp }) - H^{k}_{p}$, with $J_{1}\ne J_{2}$; so, there is at least one $a_{j_{*}}$ such that $a_{j_{*}}\ne a_{i_{l}}$ for all $l\in \{i_{1},...,i_{k-1}\}$. Then, $\langle u_{J_{1}},u_{J_{2}} \rangle _{R,S}=0$ by equation (3.3) and Proposition 3.4.

On the other hand, from Proposition 3.4 for $u_{I}=p\wedge a_{i_{1}}\wedge ...\wedge a_{i_{k}}\in H^{k}_{p}\subset \pi ^{k+1}_{\beta }(p^{\perp })$ and $u_{J}=a_{j_{1}}\wedge ...\wedge a_{j_{k+1}} \in \pi ^{k+1}_{\beta }(p^{\perp }) - H^{k}_{p}$, we obtain $\langle u_{I},u_{J} \rangle _{R,S}=0$.

From the above, Proposition 3.5 and the bilinearity of $\langle .,.\rangle _{R,S}$, we conclude that

$$\{u_{I}: I=(i_{1},...,i_{k+1}) \in I(k+1,n+1) \hspace{2mm} i_{k+1}\ne n+1\}$$

form an orthogonal basis of $\langle \pi ^{k+1}_{\beta }(p^{\perp }) \rangle$, and $\langle \pi ^{k+1}_{\beta }(p^{\perp }) \rangle \subset \langle H^{k}_{p}\rangle ^{\perp }$.

Let $x\in \langle \pi ^{k+1}_{\beta }(p^{\perp }) \rangle$ be a point, then there are $\alpha _{i} \in \mathbb {C}$ such that

$$x=\alpha _{1} u_{I_{1}}+ \cdot \cdot \cdot + \alpha _{M} u_{I_{M}},$$

where $M=\left( {\begin{array}{c}n\\ k+1\end{array}}\right)$. So,

$$\begin{aligned} \langle x,x \rangle _{R,S}= |\alpha _{1}|^{2}\langle u_{I_{1}},u_{I_{1}}\rangle _{R,S}+\cdots + |\alpha _{M}|^{2}\langle u_{I_{M}},u_{I_{M}}\rangle _{R,S} \ge 0. \end{aligned}$$

Moreover,

$$\begin{aligned} \langle x, x\rangle _{R,S}=0 \Leftrightarrow x\in \langle H^{k}_{p}\rangle . \end{aligned}$$

Hence, $\langle \pi ^{k+1}_{\beta }(p^{\perp }) \rangle \subset V^{R,S}_{0}\cup V^{R,S}_{+}$ as a vector space.

Remark 3.8

Given $u_{I}= p\wedge a_{i_{1}}\wedge \cdots \wedge a_{i_{k-1}}\wedge q \in \pi ^{k+1}_{\alpha }(p) - H^{k}_{p}$ and $u_{J}=a_{j_{1}}\wedge ...\wedge a_{j_{k+1}} \in \pi ^{k+1}_{\beta }(p^{\perp }) - H^{k}_{p}$, then $\left\langle u_{I},u_{J} \right\rangle _{R,S}=0$ by Proposition 3.4. So, from the above, Proposition 3.6 and Proposition 3.7, we conclude that

$$\begin{aligned} \{u_{I} :I=(i_{1},..., i_{k+1}) \hspace{2mm} i_{j}\ne n+1 \hspace{2mm} or \hspace{2mm} I=(1,i_{1},..., i_{k}) \} \end{aligned}$$

is an orthogonal basis of $\langle \pi ^{k+1}_{\alpha }(p),\pi ^{k+1}_{\beta }(p^{\perp })\rangle$.

Proposition 3.9

As vector spaces

$$\begin{aligned} \langle \pi ^{k+1}_{\alpha }(p),\pi ^{k+1}_{\beta }(p^{\perp })\rangle =\langle H^{k}_{p}\rangle ^{\perp }. \end{aligned}$$

Proof

From Proposition 3.6, Proposition 3.7, and the bilinearity of $\langle .,.\rangle _{R,S}$, we get that $\langle \pi ^{k+1}_{\alpha }(p),\pi ^{k+1}_{\beta }(p^{\perp })\rangle \subset \langle H^{k}_{p}\rangle ^{\perp }$.

On the one hand, $dim \langle H^{k}_{p}\rangle = \left( {\begin{array}{c}n-1\\ k\end{array}}\right)$, then $dim \langle H^{k}_{p}\rangle ^{\perp }= \left( {\begin{array}{c}n+1\\ k+1\end{array}}\right) -\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$. But, on the other hand, $dim \langle \pi ^{k+1}_{\alpha }(p),\pi ^{k+1}_{\beta }(p^{\perp })\rangle = \left( {\begin{array}{c}n\\ k\end{array}}\right) +\left( {\begin{array}{c}n\\ k+1\end{array}}\right) -\left( {\begin{array}{c}n-1\\ k\end{array}}\right) =\left( {\begin{array}{c}n+1\\ k+1\end{array}}\right) -\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$. Hence,

$$\begin{aligned} \langle \pi ^{k+1}_{\alpha }(p),\pi ^{k+1}_{\beta }(p^{\perp })\rangle = \langle H^{k}_{p}\rangle ^{\perp }. \end{aligned}$$

The above calculations extend to all $p\in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$. So, the projective subspace $\left[ \langle H^{k}_{p}\rangle \right]$ is contained in $\partial \mathbb {H}^{R,S}_{\mathbb {C}}$ for all $p\in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$, in particular for all $p\in L(\Gamma )$.

Example 3

We consider the vector space $\wedge ^{2}\mathbb {C}^{4}$, so

$$\begin{aligned} \begin{array}{rccl} P_{1,3}:&\text {Gr}_{1}(\mathbb{C}\mathbb{P}^{3})\longrightarrow & {} \mathbb {P}(\wedge ^{2}\mathbb {C}^{4}) \end{array}. \end{aligned}$$

In $\mathbb {C}^{4}$, we take the Hermitian product given by the matrix

$$\begin{aligned} H_{1,3}= \begin{pmatrix} 0&{} 0 &{} 0 &{} 1\\ 0&{} 1 &{} 0 &{} 0\\ 0&{} 0 &{} 1 &{} 0\\ 1&{} 0 &{} 0 &{} 0 \end{pmatrix}. \end{aligned}$$

Then, the induced Hermitian product in $\wedge ^{2}\mathbb {C}^{4}$ is given by the matrix $H_{3,3}= \wedge ^{2}H_{1,3}$, where

$$\begin{aligned} H_{3,3}= \begin{pmatrix} 0&{} 0 &{} 0 &{} 0 &{} -1 &{} 0\\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} -1\\ 0&{} 0 &{} -1 &{} 0 &{} 0 &{} 0\\ 0&{} 0 &{} 0 &{} 1 &{} 0 &{} 0\\ -1&{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0&{} -1 &{} 0 &{} 0 &{} 0 &{} 0 \end{pmatrix}. \end{aligned}$$

We notice that $[e_{1}] \in \partial \mathbb {H}^{1,3}_{\mathbb {C}}$ and $e_{1}^{\perp }=Span\{e_{1},e_{2},e_{3}\}$. So,

$$\begin{aligned} \begin{aligned} \left\langle \pi _{\alpha }^{2}(e_{1}) \right\rangle&=Span \{ e_{1}\wedge e_{2}, e_{1}\wedge e_{3}, e_{1}\wedge e_{4} \}, \\ \left\langle \pi _{\beta }^{2}(e_{1}^{\perp })\right\rangle&=Span\{ e_{1}\wedge e_{2}, e_{1}\wedge e_{3}, e_{2}\wedge e_{3} \},\\ \left\langle H^{2}_{e_{1}} \right\rangle&=Span \{ e_{1}\wedge e_{2}, e_{1}\wedge e_{3}\}, \\ \left\langle H^{2}_{e_{1}} \right\rangle ^{\perp }&=Span \{ e_{1}\wedge e_{2}, e_{1}\wedge e_{3}, e_{1}\wedge e_{4}, e_{2}\wedge e_{3}\} . \end{aligned} \end{aligned}$$

After calculations, we get $\langle e_{1,2},e_{1,2}\rangle _{3,3}=0= \langle e_{1,3},e_{1,3}\rangle _{3,3}=\langle e_{i,j},e_{1,4}\rangle _{3,3}= \langle e_{i,j},e_{2,3}\rangle _{3,3}$ for $(i,j)\in \{(1,2), (1,3)\}$, and $\langle e_{1,4},e_{1,4}\rangle _{3,3}=-1$, $\langle e_{2,3},e_{2,3}\rangle _{3,3}=1$. Then,

$$\begin{aligned} \left\langle \pi _{\alpha }^{2}(e_{1}) \right\rangle\subset & {} V^{R,S}_{0} \cup V^{R,S}_{-}, \\ \left\langle \pi _{\beta }^{2}(e_{1}^{\perp })\right\rangle\subset & {} V^{R,S}_{0} \cup V^{R,S}_{+},\\ \left\langle H^{2}_{e_{1}} \right\rangle\subset & {} V^{R,S}_{0}. \end{aligned}$$

4 The Limit Set

In this section, we give a candidate for a limit set of the action of $\wedge ^{k+1}\Gamma$ on $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$, where $\Gamma \subset \text {PU}(1,n)$ is a discrete subgroup.

Definition 4.1

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k < n$; we define the k-Chen-Greenberg limit set of $\Gamma$ as follows:

$$\begin{aligned} L_{k}(\Gamma ):= \overline{\bigcup _{\gamma \in Lim(\wedge ^{k+1}\Gamma )}Im(\gamma )}, \end{aligned}$$

where $\wedge ^{k+1} \Gamma \subset \text {PU}\left( \left( {\begin{array}{c}n\\ k\end{array}}\right) ,\left( {\begin{array}{c}n\\ k+1\end{array}}\right) \right)$ is a discrete subgroup acting on $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$.

From the definition, $L_{k}(\Gamma )$ is a closed set $\wedge ^{k+1}\Gamma$-invariant.

Remark 4.1

The case $k=0$, the 0-Chen-Greenberg limit set, is

$$\begin{aligned} L_{0}(\Gamma )= \overline{\bigcup _{\gamma \in Lim(\wedge ^{1}\Gamma )}Im(\gamma )}. \end{aligned}$$

On the other hand, from Lemma 4.2 and Corollary 4.3 in [5], it is just the classical limit set of Chen-Greenberg, $L(\Gamma )$.

Now, we give some properties of this limit set.

Proposition 4.2

Let $[p],[q] \in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$ be two different points, $0 \le k < n$, then $\left[ \langle H^{k}_{p} \rangle \right] \cap \left[ \langle H^{k}_{q} \rangle \right] = \emptyset$.

Proof

Let $\{a_{2},...,a_{n}\}$ be a basis of $p^{\perp } \cap q^{\perp }$; we consider $U=\{p,a_{2},...,a_{n},q\}$ an ordered basis of $\mathbb {C}^{n+1}$. This basis induces the basis $\{u_{I}: I\in I(k+1,n+1)\}$ of $\wedge ^{k+1}\mathbb {C}^{n+1}$, given by

$$\begin{aligned} u_{I}= \left\{ \begin{array}{lcc} a_{i_{1}}\wedge ... \wedge a_{i_{k+1}} &{} for &{} I=(i_{1},...,i_{k+1}) \hspace{2mm}\text{ with } \hspace{2mm} i_{j}\ne 1,n+1\\ p\wedge a_{i_{1}}\wedge ... \wedge a_{i_{k}} &{} for &{} I=(1,i_{1},...,i_{k}) \hspace{2mm}\text{ with } \hspace{2mm} i_{j}\ne n+1 \\ a_{i_{1}}\wedge ... \wedge a_{i_{k}}\wedge q &{} for &{} I=(i_{1},...,i_{k},n+1) \hspace{2mm}\text{ with } \hspace{2mm} i_{j}\ne 1 \\ p \wedge a_{i_{1}}\wedge ... \wedge a_{i_{k-1}} \wedge q &{} for &{} I=(1,i_{1},...,i_{k-1},n+1). \end{array} \right. \end{aligned}$$

So, $\left[ \langle H^{k}_{p} \rangle \right] \cap \left[ \langle H^{k}_{q} \rangle \right] = \emptyset$ for all $0 \le k < n$.

From Theorem 3.2 and Proposition 3.9, we have the following result:

Proposition 4.3

Let $(\gamma _{m})\subset \Gamma$ be a sequence tending simply to infinity, with $\Gamma \subset \text {PU}(1,n)$ a discrete subgroup, and $[\rho ], [\delta ] \in QP(\left( {\begin{array}{c}n+1\\ k+1\end{array}}\right) ,\mathbb {C})$ such that $\wedge ^{k+1}\gamma _{m}\underset{m\rightarrow \infty }{\longrightarrow }\ [\rho ]$ and $\wedge ^{k+1}\gamma _{m}^{-1} \underset{m\rightarrow \infty }{\longrightarrow }\ [\delta ]$ in the sense of pseudo-projective transformations. Then,

$$\begin{aligned} \begin{aligned} Im(\rho )&= \langle H^{k}_{p} \rangle ,\\ Im(\delta )&= \langle H^{k}_{q} \rangle ,\\ Ker(\rho )&=\langle H^{k}_{q} \rangle ^{\perp },\\ Ker(\delta )&=\langle H^{k}_{p} \rangle ^{\perp }, \end{aligned} \end{aligned}$$

for some $[p],[q] \in L(\Gamma )$. So, $Im(\rho )^{\perp }=Ker(\delta )$ and $Im(\delta )^{\perp }=Ker(\rho )$.

By Proposition 4.13 in [16] and Proposition 4.3, we have

$$\begin{aligned} \overline{\bigcup _{\gamma \in Lim(\wedge ^{k+1}\Gamma )}Im(\gamma )}=\{ [\langle H^{k}_{p} \rangle ] : p\in L(\Gamma )\}, \end{aligned}$$

so,

$$\begin{aligned} L_{k}(\Gamma )=\{ [\langle H^{k}_{p} \rangle ] : p\in L(\Gamma )\}. \end{aligned}$$

(4.4)

From Proposition 3.5 and Proposition 4.2, $L_{k}(\Gamma )$ is a collection of pairwise disjoint projective subspaces contained in $\partial \mathbb {H}^{R,S}_{\mathbb {C}}$, so, $L_{k}(\Gamma ) \subset \partial \mathbb {H}^{R,S}_{\mathbb {C}}$.

Remark 4.4

We notice that the number of projective subspaces $\left[ \langle H^{k}_{p} \rangle \right] \subset L_{k}(\Gamma )$ is determined by the cardinality of $L(\Gamma )$. So, there are 0,1,2 or infinitely many projective subspaces $\left[ \langle H^{k}_{p} \rangle \right] \subset L_{k}(\Gamma )$, and all of the same dimension i.e. of dimension $\left( {\begin{array}{c}n-1\\ k\end{array}}\right) -1$.

In this way, we say $\wedge ^{k+1}\Gamma$ is elementary if $\Gamma$ is elementary.

Proposition 4.5

Let $\Gamma \subset \text {PU}(1,n)$ be a non-elementary discrete subgroup, and $0\le k<n$, then the orbit of any projective space $[\langle H^{k}_{p} \rangle ] \subset L_{k}(\Gamma )$ is dense in $L_{k}(\Gamma )$.

Proof

Let $[p]\in L(\Gamma )$ be a point, $0\le k <n$, and $[g]\in \Gamma$ any element, then

$$\begin{aligned} \begin{aligned} \wedge ^{k+1}g(\langle H^{k}_{p} \rangle )&=\wedge ^{k+1}g ( \langle \pi ^{k+1}_{\alpha }(p)\cap \pi ^{k+1}_{\beta }(p^{\perp }) \rangle )\\&=\langle \pi ^{k+1}_{\alpha }(g(p))\cap \pi ^{k+1}_{\beta }(g(p)^{\perp }) \rangle \\&=\langle H^{k}_{g(p)} \rangle . \end{aligned} \end{aligned}$$

So,

$$\begin{aligned} \begin{aligned} \overline{Orb(\left[ \langle H^{k}_{p} \rangle \right] )}&=\overline{ \left\{ \left[ \wedge ^{k+1}g (\langle H^{k}_{p} \rangle ) \right] : [g]\in \Gamma \right\} }\\&=\overline{\left\{ \left[ \langle H^{k}_{g(p)} \rangle \right] : [g]\in \Gamma \right\} }\\&=\overline{ \left\{ \left[ \langle H^{k}_{z} \rangle \right] : [z]\in Orb([p])\right\} }\\&=\left\{ \left[ \langle H^{k}_{z} \rangle \right] : [z]\in \overline{Orb([p])} \right\} \\&=\left\{ \left[ \langle H^{k}_{z} \rangle \right] : [z]\in L(\Gamma )\right\} \\&=L_{k}(\Gamma ), \end{aligned} \end{aligned}$$

since Orb([p]) is dense in $L(\Gamma )$ by Lemma 2.1. Hence, the orbit of any projective space $[\langle H^{k}_{p} \rangle ] \subset L_{k}(\Gamma )$ is dense in $L_{k}(\Gamma )$.

For $\Gamma \subset \text {PU}(1,n)$ a discrete subgroup, $\Gamma$ acts properly discontinuously in the unitary complex ball, $\mathbb {H}^{1,n}_{\mathbb {C}}$. That does not happen in a general way; we have the following example.

Example 4

Let $[\gamma ]\in \text {PU}(1,3)$ be a loxodromic element and $[p],[q] \in \partial \mathbb {H}^{1,3}_{\mathbb {C}}$ its attracting and repelling fixed points. Consider $\Gamma = \langle [\gamma ] \rangle \subset \text {PU}(1,3)$, then $\wedge ^{2}\Gamma =\langle [\wedge ^{2} \gamma ] \rangle \subset \text {PU}(3,3)$. Now, $[\gamma _{m}]\underset{m\rightarrow \infty }{\longrightarrow }[p]$ uniformly on compact sets in $\mathbb{C}\mathbb{P}^{3} - \hspace{1mm} [q ^{\perp }]$, where $\gamma _{m}=\gamma ^{m}$.

Let $x \in \mathbb {C}^{4} - \hspace{1mm} \lbrace q ^{\perp }\cup \{ p\} \rbrace$ be a vector, so $q\wedge x\in \wedge ^{2}\mathbb {C}^{4}$, and

$$\begin{aligned} \wedge ^{2}\gamma _{m}(q\wedge x)=\gamma _{m}(q)\wedge \gamma _{m}(x)\underset{m\rightarrow \infty }{\longrightarrow }\ q \wedge p. \end{aligned}$$

From Proposition 3.4

$$\begin{aligned} \begin{aligned} \langle q\wedge x ,q\wedge x \rangle _{3,3}&=det\left( \begin{array}{rccl} \langle q,q\rangle _{1,3} &{}\langle x,q\rangle _{1,3}\\ \langle q,x\rangle _{1,3} &{}\langle x,x\rangle _{1,3} \end{array} \right) \\&= -|\langle x,q \rangle _{1,3}|^{2}<0,\\ \langle q\wedge p,q\wedge p \rangle _{3,3}&=det\left( \begin{array}{rccl} \langle q,q \rangle _{1,3} &{}\langle p,q \rangle _{1,3}\\ \langle q,p \rangle _{1,3} &{}\langle p,p\rangle _{1,3} \end{array} \right) \\&= -|\langle p,q \rangle _{1,3}|^{2} <0, \end{aligned} \end{aligned}$$

since $x,p \notin q^{\perp }$. So, $\wedge ^{2}\gamma _{m}(q\wedge x)\underset{m\rightarrow \infty }{\longrightarrow }q \wedge p$ with $[q\wedge x], [q\wedge p] \in \mathbb {H}^{3,3}_{\mathbb {C}}$. Hence, $\wedge ^{2}\Gamma$ does not act properly discontinuously in $\mathbb {H}^{3,3}_{\mathbb {C}}$.

Now, we see that this limit set can be recovered as the set of accumulation points of the orbit of some projective space contained inside $\mathbb {H}^{R,S}_{\mathbb {C}}$, that is, we look for a projective subspace $W_{}\subset \mathbb {H}^{R,S}_{\mathbb {C}}$ such that

$$\begin{aligned} L_{k}(\Gamma )=\overline{\text{ Orb }(W_{})}\cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}, \end{aligned}$$

in the same way as the Chen-Greenberg limit set: $L(\Gamma )=\overline{\text{ Orb(w) }}\cap \partial \mathbb {H}^{1,n}_{\mathbb {C}}$ for any $w\in \mathbb {H}^{1,n}_{\mathbb {C}}$.

For this, we will build the vector space $W_{k}$ in the following way. We consider $[p],[q] \in L(\Gamma )$, $[w]\in \mathbb {H}^{1,n}_{\mathbb {C}}$, and $\{a_{2},...,a_{n-1}\}$ an orthogonal basis of $w^{\perp } \cap p^{\perp }\cap q^{\perp }$, we complete with $a\in \mathbb {C}^{n+1}$ the orthogonal basis of $p^{\perp }\cap q^{\perp }$, so $p^{\perp }\cap q^{\perp }=Span\{a_{2},...,a_{n-1},a\}$, where $\langle a_{i},w \rangle _{1,n}=0$, but not necessarily $\langle a,w \rangle _{1,n}=0$ (if $w\in Span\{p,q\} \cap V^{1,n}_{-}$ then $a \in p^{\perp }\cap q^{\perp } \subset w^{\perp }$, $\langle a, w \rangle _{1,n}=0$). If $p=q$, we consider $\{a_{2},...,a_{n}\}$ an orthogonal basis of $w^{\perp } \cap p^{\perp }$. We define

$$\begin{aligned} h_{w}= \left\{ \begin{array}{lcc} Span\{w,a_{2},...,a_{n-1},a_{n}\} &{} if &{} p=q \\ Span\{w,a_{2},...,a_{n-1},a\} &{} if &{} p \ne q, \end{array} \right. \end{aligned}$$

with $h_{w} \subset \mathbb {C}^{n+1}$ a hyperplane. Now, we complete with $p\in \mathbb {C}^{n+1}$ an ordered basis of $\mathbb {C}^{n+1}$, namely,

$$\begin{aligned} U=\left\{ \begin{array}{lcc} \{w,a_{2},...,a_{n-1},a_{n},p\} &{} if &{} p=q \\ \{w,a_{2},...,a_{n-1},a,p\} &{} if &{} p \ne q. \end{array} \right. \end{aligned}$$

(4.5)

This basis is such that

$$\begin{aligned} \left\{ \begin{array}{cl} \langle p,a_{i}\rangle _{1,n}=0 =\langle w,a_{i}\rangle _{1,n} &{}\textrm{for } \; i=2,...,n\\ \langle q,a_{i}\rangle _{1,n}=0 =\langle a,a_{i}\rangle _{1,n} &{}\textrm{for } \; i=2,...,n-1\\ \langle a, p\rangle _{1,n}=0=\langle a,q \rangle _{1,n}&{}\\ \langle p, p\rangle _{1,n}=0=\langle q,q \rangle _{1,n}&{}\\ \langle a_{i},a_{j}\rangle _{1,n} \left\{ \begin{array}{cl} >0 &{}\text{ if } i=j\\ =0 &{}\text{ if } i\ne j. \end{array}\right.&\end{array}\right. \end{aligned}$$

(4.6)

We define

$$\begin{aligned} W_{k}=\langle \pi ^{k+1}_{\alpha }(w) \cap \pi ^{k+1}_{\beta }(h_{w}) \rangle , \end{aligned}$$

a vector subspace of $\wedge ^{k+1}\mathbb {C}^{n+1}$ via the Plücker embedding, where $dim(W_{k})=\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$.

Proposition 4.6

Let $W_{k}$ be the vector space built above, then $[W_{k}]\subset \mathbb {H}^{R,S}_{\mathbb {C}}$.

Proof

From the ordered basis U of $\mathbb {C}^{n+1}$ in equation (4.5), we get a basis of the vector space $W_{k}$ given by

$$\begin{aligned} W_{k}=Span\{w_{I} : I=(1,i_{1},...,i_{k}) \hspace{2mm} \text{ and } \hspace{2mm} i_{j} \ne n+1 \}, \end{aligned}$$

where

$$\begin{aligned} w_{I}= \left\{ \begin{array}{lcc} w \wedge a_{i_{1}} \wedge \cdots \wedge a_{i_{k}} &{} if &{} p=q \\ \left\{ \begin{array}{lcc} w \wedge a_{i_{1}} \wedge \cdots \wedge a_{i_{k}}\\ w \wedge a_{i_{1}} \wedge \cdots \wedge a_{i_{k-1}}\wedge a \end{array} \right.&if&p \ne q. \end{array} \right. \end{aligned}$$

Now, let us prove that $[W_{k}]\subset \mathbb {H}^{R,S}_{\mathbb {C}}$. Let $w_{I}=w\wedge a_{i_{1}} \wedge \cdots \wedge a_{i_{k}}$ and $w_{J}=w \wedge a_{j_{1}} \wedge \cdots \wedge a_{j_{k-1}}\wedge a$ be different $(k+1)-$vectors. From Proposition 3.4 and equation (4.6), we have

$$\begin{aligned} \begin{aligned} \langle w_{I},w_{I} \rangle _{R,S}=&det\left( \begin{array}{rccl} \langle w,w\rangle _{1,n} &{}\langle a_{i_{1}},w\rangle _{1,n}&{} \cdots &{} \langle a_{i_{k}},w\rangle _{1,n}\\ \langle w,a_{i_{1}}\rangle _{1,n} &{}\langle a_{i_{1}},a_{i_{1}}\rangle _{1,n}&{}\cdots &{} \langle a_{i_{k}},a_{i_{1}}\rangle _{1,n}\\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ \langle w,a_{i_{k}}\rangle _{1,n}&{} \langle a_{i_{1}},a_{i_{k}}\rangle _{1,n}&{} \cdots &{} \langle a_{i_{k}},a_{i_{k}}\rangle _{1,n} \end{array} \right) \\ =&\langle w,w \rangle _{1,n} \langle a_{i_{1}},a_{i_{1}} \rangle _{1,n} \cdots \langle a_{i_{k}},a_{i_{k}} \rangle _{1,n}<0, \\ \langle w_{J},w_{J} \rangle _{R,S}=&det\left( \begin{array}{rccl} \langle w,w\rangle _{1,n} &{}\langle a_{j_{1}},w\rangle _{1,n}&{} \cdots &{} \langle a,w\rangle _{1,n}\\ \langle w,a_{j_{1}}\rangle _{1,n} &{}\langle a_{j_{1}},a_{j_{1}}\rangle _{1,n}&{}\cdots &{} \langle a,a_{j_{1}}\rangle _{1,n}\\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ \langle w,a\rangle _{1,n}&{} \langle a_{j_{1}},a\rangle _{1,n}&{} \cdots &{} \langle a,a\rangle _{1,n} \end{array} \right) \\ =&\langle w,w \rangle _{1,n} \langle a_{j_{1}},a_{j_{1}} \rangle _{1,n} \cdots \langle a_{j_{k-1}},a_{j_{k-1}} \rangle _{1,n} \langle a,a \rangle _{1,n} \\&+ (-1)^{k+2}\langle w,a \rangle _{1,n}(-1)^{k+1}\langle a,w \rangle _{1,n}\langle a_{j_{1}},a_{j_{1}} \rangle _{1,n} \cdots \langle a_{j_{k-1}},a_{j_{k-1}} \rangle _{1,n} \\ =&\langle w,w \rangle _{1,n} \langle a_{j_{1}},a_{j_{1}} \rangle _{1,n} \cdots \langle a_{j_{k-1}},a_{j_{k-1}} \rangle _{1,n} \langle a,a \rangle _{1,n} \\&(-1) |\langle w,a \rangle _{1,n}|^{2} \langle a_{j_{1}},a_{j_{1}} \rangle _{1,n} \cdots \langle a_{j_{k-1}},a_{j_{k-1}} \rangle _{1,n}< 0, \end{aligned} \end{aligned}$$

and $\langle w_{I},w_{J} \rangle _{R,S}=0$.

On the other hand, if $w_{J_{1}}=w \wedge a_{j_{1}} \wedge \cdots \wedge a_{j_{k-1}}\wedge a$ and $w_{J_{2}}=w \wedge a_{l_{1}} \wedge \cdots \wedge a_{l_{k-1}}\wedge a$ with $J_{1} \ne J_{2}$, then $\langle w_{J_{1}},w_{J_{2}} \rangle _{R,S}=0$, since there is at least one $a_{j_{*}}$ such that $a_{j_{*}} \ne a_{l_{s}}$ for all $s\in \{1,...,k-1\}$, so

$$\begin{aligned} \langle a_{j_{*}},a_{l_{s}}\rangle _{1,n}=0=\langle a_{j_{*}},a\rangle _{1,n}=\langle a_{j_{*}},w \rangle _{1,n}. \end{aligned}$$

Analogously, for all $w_{I_{1}}=w \wedge a_{i_{1}} \wedge \cdots \wedge a_{i_{k}}$ and $w_{I_{2}}=w \wedge a_{l_{1}} \wedge \cdots \wedge a_{l_{k}}$ with $I_{1}\ne I_{2}$, we get $\langle w_{I_{1}},w_{I_{2}} \rangle _{R,S}=0$.

From the above, we conclude that

$$\begin{aligned} \{w_{I}: I=(1,i_{1},...,i_{k}) \in I(k+1,n+1) \hspace{2mm} i_{k}\ne n+1\} \end{aligned}$$

(4.7)

form an orthogonal basis of $W_{k}$. Let $x \in W_{k}-\{0\}$ be a vector, then there are $\alpha _{i} \in \mathbb {C}$ such that

$$\begin{aligned} x=\alpha _{1} w_{I_{1}}+ \cdots + \alpha _{M} w_{I_{M}}, \end{aligned}$$

where $M=dim(W_{k})=\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$. So,

$$\begin{aligned} \langle x,x \rangle _{R,S}= |\alpha _{1}|^{2}\langle w_{I_{1}},w_{I_{1}}\rangle _{R,S}+\cdots + |\alpha _{M}|^{2}\langle w_{I_{M}},w_{I_{M}}\rangle _{R,S} < 0. \end{aligned}$$

Hence, $[W_{k}] \subset \mathbb {H}^{R,S}_{\mathbb {C}}$.

Proposition 4.7

Let $[w] \in \mathbb {H}^{1,n}_{\mathbb {C}}$, $[p],[q] \in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$ be points. Then,

$$\left[ W_{k}\right] \cap \left[ \langle H^{k}_{p}\rangle \right] ^{\perp }=\emptyset =\left[ W_{k}\right] \cap \left[ \langle H^{k}_{q}\rangle \right] ^{\perp }$$

for all $0\le k< n$.

Proof

Following the notation of the previous proposition in equation (4.5), we consider

$$U=\{w,a_{2},...,a_{n-1},b,p \}$$

an ordered basis (with $b=a_{n}$ if $p=q$ and $b=a$ if $p\ne q$) of $\mathbb {C}^{n+1}$, then

$$\begin{aligned} W_{k}=Span\{w_{I} : I=(1,i_{1},...,i_{k}) \hspace{2mm} \text{ and } \hspace{2mm} i_{j} \ne n+1 \}, \end{aligned}$$

and

$$\begin{aligned} \langle H^{k}_{p}\rangle ^{\perp }= Span \{ u_{I}: I=(i_{1},...,i_{k+1}) \hspace{2mm} i_{j} \ne 1 \hspace{2mm} \text{ or } \hspace{2mm} I=(i_{1},...,i_{k},n+1)\} \end{aligned}$$

since $p^{\perp }=Span\{ a_{2},...,a_{n-1},b,p \}$. We note that $\{u_{I}, w_{I}\}$ form a basis of $\wedge ^{k+1}\mathbb {C}^{n+1}$, so $\left[ W_{k}\right] \cap \left[ \langle H^{k}_{p}\rangle \right] ^{\perp }= \emptyset$.

On the other hand, if $p\ne q$, we have $\{w,a_{2},...,a_{n-1},a,q\}$ is also an ordered basis of $\mathbb {C}^{n+1}$, where $q^{\perp }=Span\{a_{2},...,a_{n-1},a,q \}$. So, with an argument similar to the previous one we have also $\left[ W_{k}\right] \cap \left[ \langle H^{k}_{q}\rangle \right] ^{\perp }= \emptyset$.

Proposition 4.8

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0 \le k<n$, then

$$\begin{aligned} L_{k}(\Gamma )= \overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}. \end{aligned}$$

Proof

Recall,

$$\begin{aligned} L_{k}(\Gamma )=\{ [\langle H^{k}_{p}\rangle ]: [p]\in L(\Gamma ) \}. \end{aligned}$$

If $\Gamma$ is a finite group, then $L(\Gamma )=\emptyset$, and $\wedge ^{k+1}\Gamma$ is also a finite group, so $L_{k}(\Gamma )=\emptyset$. If $\Gamma$ is not a finite group, we separate it by cases depending on the cardinality of $L(\Gamma )$.

Case 1: If $L(\Gamma )=\{[p]\}$, there is $[\gamma ]\in \Gamma$ parabolic element with fixed point [p]. We consider $\gamma _{m}=\gamma ^{m}$, from Proposition 2.2, there is a subsequence of $(\gamma _{m})$, still denoted $(\gamma _{m})$, and $\tau \in QP(n+1,\mathbb {C})$ such that $[\gamma _{m}]\underset{m\rightarrow \infty }{\longrightarrow }\tau$ uniformly on compact sets of $\mathbb{C}\mathbb{P}^{n} - \hspace{1 mm}[p^{\perp }]$ where $Im(\tau )=[p]$.

Now, from Theorem 3.2, there is $\rho \in QP(N,\mathbb {C})$ such that $[\wedge ^{k+1}\gamma _{m}]\underset{m\rightarrow \infty }{\longrightarrow }\rho$ uniformly on compact sets of $\mathbb{C}\mathbb{P}^{N-1} - \hspace{1 mm} [\langle H^{k}_{p}\rangle ]^{\perp }$, where $Im(\rho )=[\langle H^{k}_{p}\rangle ]$.

From Proposition 4.7, we have built $W_{k}$ in such a way that $[W_{k}] \subset \mathbb{C}\mathbb{P}^{N-1} - \hspace{1mm} [\langle H^{k}_{p}\rangle ]^{\perp }$; so $\wedge ^{k+1}\gamma _{m}(W_{k})\underset{m\rightarrow \infty }{\longrightarrow }\ \langle H^{k}_{p}\rangle$ since $dim (W_{k})=M=dim \langle H^{k}_{p}\rangle$. Hence,

$$\begin{aligned} L_{k}(\Gamma )=\{[\langle H^{k}_{p}\rangle ]\} \subset \overline{ Orb ([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}. \end{aligned}$$

Case 2: If $L(\Gamma )=\{[p],[q]\}$, there is $[\gamma ]\in \Gamma$ loxodromic element with fixed points [p] and [q]. We consider $\gamma _{m}=\gamma ^{m}$, from Proposition 2.2 there is a subsequence of $(\gamma _{m})$, still denoted $(\gamma _{m})$, and $\tau , \theta \in QP(n+1,\mathbb {C})$ such that $[\gamma _{m}]\underset{m\rightarrow \infty }{\longrightarrow }\tau$, $[\gamma ^{-1}_{m}]\underset{m\rightarrow \infty }{\longrightarrow }\theta$ uniformly on compact sets of $\mathbb{C}\mathbb{P}^{n} - Ker(\tau )$ and $\mathbb{C}\mathbb{P}^{n} - Ker(\theta )$ respectively; where $Ker(\tau )=[p^{\perp }], Ker(\theta )=[q^{\perp }], Im(\tau )=[q]$ and $Im(\theta )=[p]$.

Now, from Theorem 3.2, there exist $\rho , \delta \in QP(N,\mathbb {C})$ such that $[\wedge ^{k+1} \gamma _{m}]\underset{m\rightarrow \infty }{\longrightarrow }\rho$, $[\wedge ^{k+1} \gamma ^{-1}_{m}]\underset{m\rightarrow \infty }{\longrightarrow }\delta$ uniformly on compact sets of $\mathbb{C}\mathbb{P}^{N-1}- Ker(\rho )$ and $\mathbb{C}\mathbb{P}^{N-1}-Ker(\delta )$ respectively; where $Ker(\rho )=[\langle H^{k}_{p} \rangle ]^{\perp }$, $Ker(\delta )=[\langle H^{k}_{q} \rangle ]^{\perp }$, $Im(\rho )=[\langle H^{k}_{q} \rangle ]$, and $Im(\delta )=[\langle H^{k}_{p} \rangle ]$.

From Proposition 4.7, we have built the set $[W_{k}]$ in such a way that $[W_{k}] \subset \mathbb{C}\mathbb{P}^{N-1} - \hspace{1mm} [\langle H^{k}_{p}\rangle ] ^{\perp }$ and $[W_{k}] \subset \mathbb{C}\mathbb{P}^{N-1} - \hspace{1mm} [\langle H^{k}_{q}\rangle ] ^{\perp }$, thence $\wedge ^{k+1}\gamma _{m}(W_{k})\underset{m\rightarrow \infty }{\longrightarrow }\ \langle H^{k}_{q}\rangle$ and $\wedge ^{k+1}\gamma ^{-1}_{m}(W_{k})\underset{m\rightarrow \infty }{\longrightarrow }\langle H^{k}_{p}\rangle$ since $dim (W_{k})=M=dim \langle H^{k}_{p}\rangle =dim \langle H^{k}_{q}\rangle$. So,

$$\begin{aligned} L_{k}(\Gamma )=\{[\langle H^{k}_{p}\rangle ], [\langle H^{k}_{q}\rangle ]\} \subset \overline{ Orb ([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}. \end{aligned}$$

Case 3: If $|L(\Gamma )|>2$, then $\Gamma$ is non-elementary. From Proposition 2.1, there is $[\gamma ]\in \Gamma$ loxodromic element with fixed point [p]. With an argument similar to the previous one, we have

$$\begin{aligned}{}[\langle H^{k}_{p} \rangle ]\subset \overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}. \end{aligned}$$

Hence,

$$\begin{aligned} L_{k}(\Gamma )= \overline{Orb ([\langle H^{k}_{p} \rangle ]) } \subset \overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}\end{aligned}$$

since $\overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}$ is a closed $\wedge ^{k+1}\Gamma$-invariant set and by Proposition 4.5.

From the above, we conclude that $L_{k}(\Gamma ) \subset \overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}$. Let us now prove the other inclusion

$$\begin{aligned} \overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}\subset L_{k}(\Gamma ). \end{aligned}$$

Let $[V]\subset \partial \mathbb {H}^{R,S}_{\mathbb {C}}$ be a projective space such that $\wedge ^{k+1}\gamma _{m}(W_{k}) \underset{m\rightarrow \infty }{\longrightarrow }\ V$ where $([\gamma _{m}])\subset \Gamma$ is a sequence of distinct elements, so $dim(V)=dim (W_{k})=M$. From equality in (4.7), we get $\{w_{I}\}$ is an orthogonal basis of $W_{k}$, with $[w_{I}]\in Q_{k,n}$; so there are $\{V_{1},...,V_{M}\}\subset \wedge ^{k+1}\mathbb {C}^{n+1}$ linearly independent vectors such that $V=Span\{V_{1},...,V_{M}\}$ with $[V_{i}]\in Q_{k,n}$, and $\wedge ^{k+1}\gamma _{m}(w_{I_{i}}) \underset{m\rightarrow \infty }{\longrightarrow }V _{i}$; so there are vectors $\{v_{i}\}\in \mathbb {C}^{n+1}$ such that $V_{i}=v_{i_{1}}\wedge ...\wedge v_{i_{k+1}}$ for each $i\in \{1,...,M\}$.

On the other hand, recall, $[w]\in \mathbb {H}^{1,n}_{\mathbb {C}}$, $\{a_{i_{j}}\} \in w^{\perp }$ and $a \in p^{\perp }\cap q^{\perp }$, so

$$\begin{aligned} \begin{aligned} \gamma _{m}(w)&\underset{m\rightarrow \infty }{\longrightarrow }v \hspace{3mm} \text{ with } \hspace{2mm} [v]\in L(\Gamma )\\ \gamma _{m}(a_{i_{j}})&\underset{m\rightarrow \infty }{\longrightarrow }v_{i_{j}} \in v^{\perp } \hspace{3mm} \text{ for } \hspace{2mm} 2 \le i_{j} \le n\\ \gamma _{m}(a)&\underset{m\rightarrow \infty }{\longrightarrow }v_{0}. \end{aligned} \end{aligned}$$

Now, since $\langle a_{i}, a_{j} \rangle _{1,n}=\langle a, a_{i}\rangle _{1,n}= \langle a_{i},w \rangle _{1,n}=0$ for all $i\ne j$, then $\langle v, v_{i} \rangle _{1,n}= \langle v_{i}, v_{j}\rangle _{1,n}=0$ for all $i\ne j$ and $\langle v_{i},v_{i}\rangle _{1,n} >0$.

Hence, for each $w_{I}\in W_{k}$:

$$\begin{aligned} \wedge ^{k+1}\gamma _{m}(w_{I})= \left\{ \begin{array}{lcc} \gamma _{m}(w)\wedge \gamma _{m}(a_{i_{1}})\wedge ... \wedge \gamma _{m}(a_{i_{k-1}}) \wedge \gamma _{m}(a_{i_{k}})\\ \gamma _{m}(w)\wedge \gamma _{m}(a_{i_{1}})\wedge ... \wedge \gamma _{m}(a_{i_{k-1}})\wedge \gamma _{m}(a) \end{array} \right. \end{aligned}$$

is such that

$$\begin{aligned} \begin{aligned} \gamma _{m}(w)\wedge \gamma _{m}(a_{i_{1}})\wedge ... \wedge \gamma _{m}(a_{i_{k-1}}) \wedge \gamma _{m}(a_{i_{k}})&\underset{m\rightarrow \infty }{\longrightarrow }\ v \wedge v_{i_{1}}\wedge ...\wedge v_{i_{k-1}}\wedge v_{i_{k}},\\ \gamma _{m}(w)\wedge \gamma _{m}(a_{i_{1}})\wedge ... \wedge \gamma _{m}(a_{i_{k-1}})\wedge \gamma _{m}(a)&\underset{m\rightarrow \infty }{\longrightarrow }v \wedge v_{i_{1}}\wedge ...\wedge v_{i_{k-1}}\wedge v_{0}. \end{aligned} \end{aligned}$$

If $\langle a,w \rangle _{1,n}=0$ then $\langle v, v_{0}\rangle _{1,n}=0$, so $v_{0} \in v^{\perp }$. We prove that even if $\langle a,w \rangle _{1,n} \ne 0$, $v_{0} \in v ^{\perp }$. Given $V_{i}=v\wedge v_{i_{1}}\wedge \cdots \wedge v_{i_{k-1}}\wedge v_{0} \in V$, $[V_{i}]\in \partial \mathbb {H}^{R,S}_{\mathbb {C}}$, from Proposition 3.4, we get

$$\begin{aligned} \begin{aligned} 0&=\langle V_{i},V_{i} \rangle _{R,S}\\&=det\left( \begin{array}{rccl} \langle v,v\rangle _{1,n} &{}\langle v_{i_{1}},v\rangle _{1,n}&{} \cdots &{} \langle v_{0},v\rangle _{1,n}\\ \langle v,v_{i_{1}}\rangle _{1,n} &{}\langle v_{i_{1}},v_{i_{1}}\rangle _{1,n}&{}\cdots &{} \langle v_{0},v_{i_{1}}\rangle _{1,n}\\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ \langle v,v_{0}\rangle _{1,n}&{} \langle v_{i_{1}},v_{0}\rangle _{1,n}&{} \cdots &{} \langle v_{0},v_{0}\rangle _{1,n} \end{array} \right) \\&= (-1)^{k+2}\langle v,v_{0} \rangle _{1,n}(-1)^{k+1} \langle v_{0},v \rangle _{1,n}\langle v_{i_{1}},v_{i_{1}} \rangle _{1,n} \cdots \langle v_{i_{k-1}},v_{i_{k-1}} \rangle _{1,n}\\&= (-1) |\langle v,v_{0} \rangle _{1,n}|^{2} \langle v_{i_{1}},v_{i_{1}} \rangle _{1,n} \cdots \langle v_{i_{k-1}},v_{i_{k-1}} \rangle _{1,n}=0, \end{aligned} \end{aligned}$$

then $\langle v,v_{0} \rangle _{1,n}=0$, so $v_{0}\in v^{\perp }$.

From the above and the bilinearity of $\langle .,. \rangle _{R,S}$, we deduce that:

• (1) If $a\notin w_{I_{j}}$ then $V_{j}=v\wedge v_{i_{1}}\wedge ...\wedge v_{i_{k-1}}\wedge v_{i_{k}}$ is such that $v \in V_{j} \subset v^{\perp }$, since $v_{i_{l}}\in v^{\perp }$.

• (2) If $a \in w_{I_{j}}$ then $V_{j}=v\wedge v_{i_{1}}\wedge ...\wedge v_{i_{k-1}}\wedge v_{0}$ is such that $v \in V_{j} \subset v^{\perp }$ too, since $v_{i_{j}}\in v^{\perp }$ and also $v_{0}\in v^{\perp }$.

So, $v \in V_{j} \subset v^{\perp }$ for all $j\in \{1,...,M\}$, where $\{V_{1},...,V_{M}\}$ are M linearly independent vectors, thus $[V]=[\langle H^{k}_{v} \rangle ] \subset L_{k}(\Gamma )$. Hence, $\overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}\subset L_{k}(\Gamma )$.

Therefore,

$$\begin{aligned} L_{k}(\Gamma )= \overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}. \end{aligned}$$

Summarizing, we have the following:

Theorem 4.9

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup and let $L_{k}(\Gamma )=\{ [\langle H^{k}_{p} \rangle ]: [p]\in L(\Gamma )\}.$ Then,

• (1) $L_{k}(\Gamma )$ has 0,1,2 or infinitely many projective subspaces $[\langle H^{k}_{p} \rangle ]$; and all of the same dimension..

• (2) For $0 \le k <n$, $[W_{k}]\subset \mathbb {H}^{R,S}_{\mathbb {C}}$.

• (3) $L_{k}(\Gamma )= \overline{ Orb([W_{k}])} \cap \partial \mathbb {H}^{R,S}_{\mathbb {C}}.$

• (4) If $\Gamma$ is a non-elementary discrete group then the orbit of any projective space $[\langle H^{k}_{p} \rangle ] \subset L_{k}(\Gamma )$ is dense in $L_{k}(\Gamma )$.

From Proposition 3.9, we can rewrite the Theorem 4.14 in [16] as follows:

Corollary 4.10

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0 \le k <n$, then

$$\begin{aligned} \mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1}) - \hspace{1mm} Eq(\wedge ^{k+1}\Gamma )=\bigcup _{[ \langle H^{k}_{p} \rangle ] \subset L_{k}(\Gamma )} \left[ \langle H^{k}_{p} \rangle \right] ^{\perp } \end{aligned}$$

where $[p]\in L(\Gamma )$ and $L_{k}(\Gamma )$ is the k-Chen-Greenberg limit set.

5 Kulkarni Limit Set

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k<n$; in this section, we determine the Kulkarni limit set of the discrete subgroup $\wedge ^{k+1}\Gamma \subset \text {PU}\left( \left( {\begin{array}{c}n\\ k\end{array}}\right) ,\left( {\begin{array}{c}n\\ k+1\end{array}}\right) \right)$ acting on $\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$.

We extend the idea used in [2] to our case; for this, we need a tool that enables us to determine the accumulation points of the orbit of compact sets (see [8, 11]).

Definition 5.1

Let $(\gamma _{m}) \subset \text {PU}(r,s)$ be a sequence tending simply to infinity, and $x\in \mathbb{C}\mathbb{P}^{r+s-1}$. We define

$$ \mathcal {D}_{(\gamma _{m})}(x)= \bigcup \{\text{ accumulation } \text{ points } \text{ of } \hspace{2mm} (\gamma _{m}(x_{m}))\}.$$

The union is taken over all sequences converging to x.

If $\Gamma \subset \text {PU}(r,s)$ is a discrete group, $\Omega \subset \mathbb{C}\mathbb{P}^{r+s-1}$ is an open set on which $\Gamma$ acts properly discontinuously, $x\in \Omega$, and $(\gamma _{m}) \subset \Gamma$ is any sequence tending simply to infinity, then $\mathcal {D}_{(\gamma _{m})}(x)\subset \mathbb{C}\mathbb{P}^{r+s-1} - \hspace{1mm} \Omega$.

Definition 5.2

Let $D \subset \mathbb{C}\mathbb{P}^{N-1}$ be a projective space such that $D \subset D^{\perp }$ and $dim(D)=M-1$, we define

$$\begin{aligned} D^{\perp }(D)=\{ \ell \in \text {Gr}_{M}(\mathbb{C}\mathbb{P}^{N-1}): D\subset \ell \subset D^{\perp } \}. \end{aligned}$$

In our case, to simplify the notation, given $0\le k<n$ and $p\in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$, we define

$$\begin{aligned} \mathcal {E}^{k}_{p} :=\{ \ell \in \text {Gr}_{M}(\mathbb{C}\mathbb{P}^{N-1}): [\langle H^{k}_{p} \rangle ] \subset \ell \subset [\langle H^{k}_{p} \rangle ]^{\perp }\} \end{aligned}$$

where $M=\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$.

The following proposition is straightforward.

Lemma 5.1

Given $0\le k<n$ and $p\in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$, define the following function in $\mathcal {E}^{k}_{p} \times \mathcal {E}^{k}_{p}$:

$$\begin{aligned} d_{p}(\ell _{1},\ell _{2})=arccos\left( \sqrt{\dfrac{\langle q_{1},q_{2} \rangle _{R,S}\langle q_{1},q_{2} \rangle _{R,S}}{\langle q_{1},q_{1} \rangle _{R,S} \langle q_{2},q_{2} \rangle _{R,S}}} \right) , \end{aligned}$$

where $q_{1},q_{2} \in \mathbb {C}^{n+1} - \{0\}$ are points satisfying $\ell _{i}= [Span\{H^{k}_{p},q_{i}\}]$. In this way, we obtain what $(\mathcal {E}^{k}_{p},d_{p})$ is a metric space isometric to $\left( \mathbb{C}\mathbb{P}^{N-2M},d_{N-2M}\right)$.

Definition 5.3

Given $p,q\in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$, we will denote by $Isom_{k}(p,q)$ the set of isometries from $(\mathcal {E}^{k}_{p},d_{p})$ to the space $(\mathcal {E}^{k}_{q},d_{q})$.

Remark 5.2

Given the Hermitian matrix $H_{1,n}$ then

$$\begin{aligned} \begin{aligned} \wedge ^{k+1}H_{1,n}(e_{I})&= H_{1,n}(e_{i_{1}})\wedge ...\wedge H_{1,n}(e_{i_{k+1}})\\&=\left\{ \begin{array}{lcc} e_{n+1}\wedge e_{i_{2}}\wedge ...\wedge e_{i_{k+1}} \in B &{} if &{} I=(1,i_{2},...,i_{k+1})\in A\\ e_{i_{1}}\wedge ...\wedge e_{i_{k}}\wedge e_{1} \in A &{} if &{} I=(i_{1},...,i_{k},n+1)\in B\\ e_{i_{1}}\wedge ...\wedge e_{i_{k+1}}\in A^{c}\cap B^{c} &{} if &{} I=(i_{1},...,i_{k+1})\in A^{c}\cap B^{c}, \end{array} \right. \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} A&=\{I=(1,i_{2},...,i_{k+1}) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne n+1\},\\ B&=\{I=(i_{1},...,i_{k},n+1) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne 1\}. \end{aligned} \end{aligned}$$

The following proposition extends the result in [2, Proposition 3.6] in some way.

Proposition 5.3

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k<n$, let $(\gamma _{m})\subset \Gamma$ be a sequence tending simply to infinity, and let $\rho , \delta \in QP(N,\mathbb {C})$ be pseudo-projective transformations such that $\wedge ^{k+1}\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \rho$ and $\wedge ^{k+1}\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\delta$. Then, there is a projective equivalence

$$\begin{aligned} \varphi _{k}: Im(\delta )^{\perp }(Im(\delta )) \longrightarrow Im(\rho )^{\perp }(Im(\rho )) \end{aligned}$$

satisfying the following:

• (1) The equivalence $\varphi _{k}$ belongs to $Isom_{k}(q,p)$ where q and p are points such that $Im(\delta )=[\langle H^{k}_{q} \rangle ]$ and $Im(\rho )=[\langle H^{k}_{p} \rangle ]$.

• (2) Given $\ell \in Im(\delta )^{\perp }(Im(\delta ))$, and $x\in \ell \hspace{1mm} - Im(\delta )$, we know

  $$\begin{aligned} \mathcal {D}_{(\wedge ^{k+1}\gamma _{m})}(x)=\varphi _{k}(\ell ). \end{aligned}$$

• (3) Given $\ell \in Im(\rho )^{\perp }(Im(\rho ))$, and $y\in \ell \hspace{1mm} -Im(\rho )$, we know

  $$\begin{aligned} \mathcal {D}_{(\wedge ^{k+1}\gamma ^{-1}_{m})}(y)=\varphi _{k}^{-1}(\ell ). \end{aligned}$$

Proof

Let $(\gamma _{m})\subset \Gamma$ be a sequence tending simply to infinity. By the Cartan decomposition theorem, there are sequences $(\lambda _{m})\in \mathbb {R}^{+}, (g^{}_{m}), (h^{}_{m}) \in K=\text {U}(1,n)\cap \text {U}(1+n)$, such that $\gamma _{m}=\left[ g^{}_{m}\mu (\gamma _{m})h^{}_{m}\right]$, where

$$\begin{aligned} \mu (\gamma _{m})= \begin{pmatrix} e^{\lambda _{m}}&{} &{} &{} &{} \\ &{} 1 &{} &{} &{}\\ &{} &{} \ddots &{} &{}\\ &{} &{} &{} 1 &{} \\ &{} &{} &{} &{} e^{-\lambda _{m}} \end{pmatrix}. \end{aligned}$$

We can assume that

$$\begin{aligned} g_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ g \in K, \hspace{2mm} \hspace{2mm} h_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ h\in K\hspace{2mm} \text{ and }\hspace{2mm} \lambda _{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \infty . \end{aligned}$$

Now, $\wedge ^{k+1}\gamma _{m}=\left[ \wedge ^{k+1}g^{}_{m}\wedge ^{k+1}\mu (\gamma _{m})\wedge ^{k+1}h^{}_{m}\right]$ for $0\le k < n$. From Theorem 3.2, there are $\rho , \delta \in QP(N,\mathbb {C})$ such that

$$\begin{aligned} \wedge ^{k+1}\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \rho \hspace{2mm} \text{ and } \hspace{2mm} \wedge ^{k+1}\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \delta \end{aligned}$$

in the sense of pseudo-projective transformations. From Proposition 4.3, the sets $Im(\rho )$ and $Im(\delta )$ are contained in the k-Chen-Greenberg limit set and

$$\begin{aligned} Im(\delta )^{\perp }= Ker(\rho ) \hspace{2mm} \text{ and } \hspace{2mm} Im(\rho )^{\perp }= Ker(\delta ), \end{aligned}$$

where $dim(Im(\rho ))=dim(Im(\delta ))=\left( {\begin{array}{c}n-1\\ k\end{array}}\right) -1=M-1$ and $dim(Ker(\rho ))=dim(Ker(\delta ))=N-M-1$.

We consider the projective equivalence

$$\begin{aligned} \varphi _{k}: Im(\delta )^{\perp }(Im(\delta )) \longrightarrow Im(\rho )^{\perp }(Im(\rho )) \end{aligned}$$

defined by $\varphi _{k}=\left[ \wedge ^{k+1}g_{}\wedge ^{k+1}H_{1,n}\wedge ^{k+1}h_{}\right] \in \text {PU}(R,S)$.

First, let us see that $\varphi _{k}$ is well defined: given $\ell \in \text {Gr}_{M}(\mathbb{C}\mathbb{P}^{N-1})$ such that $Im(\delta ) \subset \ell \subset Im(\delta )^{\perp }$, from equalities in (3.1) and Remark 5.2, we have

$$\begin{aligned} \begin{aligned} Im(\rho )&= \wedge ^{k+1}g_{}\wedge ^{k+1}H_{1,n}\wedge ^{k+1}h_{} (Im(\delta ))\\ Im(\rho )^{\perp }&= \wedge ^{k+1}g_{}\wedge ^{k+1}H_{1,n}\wedge ^{k+1}h_{} (Im(\delta )^{\perp }) , \end{aligned} \end{aligned}$$

then $Im(\rho ) \subset \varphi _{k}(\ell ) \subset Im(\rho )^{\perp }$, so $\varphi _{k}$ is well defined.

It is not hard to see that $\varphi _{k}$ is an isometry from $\left( Im(\delta )^{\perp }(Im(\delta )),d_{q}\right)$ to the space $\left( Im(\rho )^{\perp }(Im(\rho )),d_{p}\right)$, where q and p are points such that $Im(\delta )=[\langle H^{k}_{q} \rangle ]$ and $Im(\rho )=[\langle H^{k}_{p} \rangle ]$.

Now, we show part 2). Consider $\ell \in Im(\delta )^{\perp }(Im(\delta ))$, $x\in \ell - \hspace{1mm} Im(\delta )$, and $\left( x_{m}\right) \subset \mathbb{C}\mathbb{P}^{N-1}$ such that $x_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ x$, then $\wedge ^{k+1}h_{}(x_{m}) \underset{m\rightarrow \infty }{\longrightarrow }\ \wedge ^{k+1}h_{}(x)$; from the equalities in (3.1), we have $\wedge ^{k+1}h_{}(x) \in Span\{e_{I}:I\notin A\} - Span\{e_{I}: I\in B \}$. So, $\wedge ^{k+1}h_{}(x_{m})=(x_{I_{1}m},...,x_{I_{N}m})$, $\wedge ^{k+1}h_{}(x)=(x_{I_{1}},...,x_{I_{N}})$, where $x_{I}=0$ for all $I\in A$, $\sum _{I\in A^{c}\cap B^{c}}|x_{I}| \ne 0$, and $x_{I_{j}m} \underset{m\rightarrow \infty }{\longrightarrow }\ x_{I_{j}}$ for all $j\in \{1,...,N \}$; where

$$\begin{aligned} \begin{aligned} A&=\{I=(1,i_{2},...,i_{k+1}) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne n+1\},\\ B&=\{I=(i_{1},...,i_{k},n+1) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne 1\}. \end{aligned} \end{aligned}$$

Now,

$$\begin{aligned} (\wedge ^{k+1}\mu (\gamma _{m})) (e_{I})= \left\{ \begin{array}{lcc} e^{\lambda _{m}}e_{I} &{} if &{} I \in A\\ e_{I} &{} if &{} I \in A^{c}\cap B^{c}\\ e^{-\lambda _{m}}e_{I} &{} if &{} I \in B \end{array} \right. \end{aligned}$$

We define, $y_{m}=\wedge ^{k+1}\mu (\gamma _{m}) \left( \begin{array}{rccl} x_{I_{1}m}\\ \vdots \hspace{3mm}\\ x_{I_{N}m} \end{array} \right)$, so,

$$\begin{aligned} y_{m}^{}= \sum _{I\in A} e^{\lambda _{m}}x_{Im}e_{I} + \sum _{I\in A^{c}\cap B^{c}} x_{Im}e_{I} + \sum _{I\in B} e^{-\lambda _{m}}x_{Im}e_{I}; \end{aligned}$$

(5.8)

thus the accumulation points of the sequence $(y_{m})$ lie on the vector space

$$\begin{aligned} Span\{Span\{e_{I}:I\in A\}, y\}, \end{aligned}$$

where $y=\sum _{I\in A^{c}\cap B^{c}}^{}x_{I}e_{I}$. Hence, the accumulation points of $(\wedge ^{k+1}\gamma _{m}(x_{m}))$ lie on the vector space

$$\begin{aligned} Span\{\wedge ^{k+1}g_{}(Span\{e_{I}:I\in A\}), \wedge ^{k+1}g_{}(y)\}, \end{aligned}$$

so, on the vector space $Span\{Im(\rho ), \wedge ^{k+1}g_{}(y)\}$.

On the other hand, from the equalities in (3.1) and the equation (5.8), we know that $\varphi _{k}(\ell )= [Span\{Im(\rho ), \wedge ^{k+1}g_{}(y)\}]$. Thus, $\mathcal {D}_{(\wedge ^{k+1}\gamma _{m})}(x)\subset \varphi _{k}(\ell )$. Now, from the above and the Proposition 3.4 in [1], we conclude that

$$\begin{aligned} \mathcal {D}_{(\wedge ^{k+1}\gamma _{m})}(x)=\varphi _{k}(\ell ). \end{aligned}$$

In an analogous way, we have the part 3).

Inspired by the previous proposition, we have the next definition.

Definition 5.4

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete group, $0\le k <n$ and $(\gamma _{m})\subset \Gamma$ be a sequence tending simply to infinity whose Cartan decomposition is given by $\gamma _{m}=g^{}_{m}\mu (\gamma _{m})h^{}_{m}$, where $g^{}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ g_ {}$ and $h^{}_{m}\underset{m\rightarrow \infty }{\longrightarrow }h_ {}$, with $g,h \in \text {PU}(1,n)\cap \text {PU}(1+n)$. The transformation

$$\begin{aligned} \varphi _{k}: Im(\delta )^{\perp }(Im(\delta ))\longrightarrow Im(\rho )^{\perp }(Im(\rho )) \end{aligned}$$

defined by $\varphi _{k}=\wedge ^{k+1}g_{}\wedge ^{k+1}H_{1,n}\wedge ^{k+1}h_{}$ is called a k-Cartan map associated with $(\gamma _{m})$, where $\wedge ^{k+1}\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \rho$ and $\wedge ^{k+1}\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\delta$ in the sense of pseudo-projective transformations.

Definition 5.5

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete group, $0\le k< n$, $p,q\in L(\Gamma )$. We define

$$\begin{aligned} \Gamma _{k}(p,q)=\{ \varphi _{k}: \mathcal {E}^{k}_{p} \longrightarrow \mathcal {E}^{k}_{q} \hspace{1mm}| \hspace{1mm} \varphi _{k} \text{ is } \text{ a } \text{ k-Cartan } \text{ map } \}, \end{aligned}$$

where $\varphi _{k}$ is associated with $(\gamma _{m})$, sequence tending simply to infinity.

We extend the definition of the binary operation $\star$ in [2] to our case, $Bihol_{k}(p,q)$.

Definition 5.6

Let $p,q \in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$, $0\le k< n$, and let

$$\begin{aligned} Bihol_{k}(p,q)=\{ \varphi _{k}:\mathcal {E}^{k}_{p} \longrightarrow \mathcal {E}^{k}_{q}: \varphi _{k} \text{ is } \text{ biholomorphic } \}. \end{aligned}$$

We define the binary operation $\star$ on $Bihol_{k}(p,q)$:

$$\begin{aligned} \mu \star \nu (\ell )= \left\{ \begin{array}{lcc} \mu (\nu (\ell )) &{} if &{} p=q,\\ \mu \left( Span \left\{ \nu (\ell )\cap \mathcal {E}^{k}_{p}, [\langle H^{k}_{p} \rangle ] \right\} \right) &{} &{} otherwise. \end{array} \right. \end{aligned}$$

The operation $\star$ is quite usual even in our case.

Lemma 5.4

Given $p,q \in \partial \mathbb {H}^{1,n}_{\mathbb {C}}$, $0\le k< n$, $(Bihol_{k}(p,q), \star )$ is a group isomorphic to $(\text {PSL}(N-2M,\mathbb {C}), \circ )$. Moreover, $(Isom_{k}(p,q),\star$) is a group isomorphic to $(\text {PU}(N-2M),\circ )$.

We need the following result:

Lemma 5.5

(Lemma 4.13 in [2]) Let $\Gamma \subset \text {PU}(1,n)$ be a non-elementary discrete subgroup and $p\in L(\Gamma )$. Then,

$$\begin{aligned} \Gamma (p,p)=\{ \varphi : p^{\perp }(p) \longrightarrow p^{\perp }(p) \hspace{1mm}| \hspace{1mm} \varphi \text{ is } \text{ a } \text{ Cartan } \text{ map } \} \end{aligned}$$

contains the identity.

Let us observe that the result in [2] corresponds to the case $k = 0$, namely $\Gamma _{0}(p,q)=\Gamma (p,q)$.

Corollary 5.6

Let $\Gamma \subset \text {PU}(1,n)$ be a non-elementary discrete subgroup, $p\in L(\Gamma )$. Then, $id\arrowvert _{\mathcal {E}^{k}_{p}}\in \Gamma _{k}(p,p)$ for all $0\le k< n$.

Proof

From Lemma 5.5, we know $id_{p^{\perp }(p)} \in \Gamma _{0}(p,p)$, so there is $(\gamma _{m})\subset \Gamma$ a sequence tending simply to infinity whose Cartan decomposition is giving by $\gamma _{m}=g^{}_{m}\mu (\gamma _{m})h^{}_{m}$, where $g^{}_{m} \underset{m\rightarrow \infty }{\longrightarrow }g_ {}$ and $h^{}_{m} \underset{m\rightarrow \infty }{\longrightarrow }h_ {}$, with $g, h \in \text {PU}(1,n)\cap \text {PU}(1+n)$, such that $\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \theta$ and $\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \tau$ uniformly on compact sets in $\mathbb{C}\mathbb{P}^{n} - \hspace{1mm} p^{\perp }$, where $Im(\theta )=Im(\tau )=\left\{ p \right\} \in L(\Gamma )$. So,

$$\begin{aligned} id:p^{\perp }(p)\longrightarrow p^{\perp }(p) \end{aligned}$$

defined by $id_{p^{\perp }(p)}=g_{}H_{1,n}h_{}\arrowvert _{p^{\perp }(p)}$ is the Cartan map associated with $(\gamma _{m})$.

Now, from Theorem 3.2, we have $\wedge ^{k+1}\gamma _{m}=\wedge ^{k+1}g^{}_{m}\wedge ^{k+1}\mu (\gamma _{m})\wedge ^{k+1}h^{}_{m}$, where $\wedge ^{k+1}g^{}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \wedge ^{k+1}g_{}$ and $\wedge ^{k+1}h^{}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\wedge ^{k+1}h_{}$, with $\wedge ^{k+1}g_{}, \wedge ^{k+1}h_{} \in \text {PU}(R,S)$; and $\rho , \delta \in QP(N,\mathbb {C})$ such that $\wedge ^{k+1}\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \rho$ and $\wedge ^{k+1}\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ \delta$ uniformly on compact sets in $\mathbb{C}\mathbb{P}^{N-1} - \hspace{0mm} [\langle H^{k}_{p} \rangle ]^{\perp }$, with $Im(\rho )=Im(\delta )=[\langle H^{k}_{p}\rangle ] \subset L_{k}(\Gamma )$. Hence,

$$\begin{aligned} \begin{aligned} \wedge ^{k+1}id&=\wedge ^{k+1}g_{}\wedge ^{k+1}H_{1,n}\wedge ^{k+1} h_{}\\&=\wedge ^{k+1}g_{}H_{R,S}\wedge ^{k+1} h_{} \end{aligned} \end{aligned}$$

is the k-Cartan map associated with $(\gamma _{m})$. So,

$$\begin{aligned} \wedge ^{k+1}id:\mathcal {E}^{k}_{p}\longrightarrow \mathcal {E}^{k}_{p} \end{aligned}$$

defined by $\wedge ^{k+1}g_{}H_{R,S}\wedge ^{k+1} h_{}\arrowvert _{\mathcal {E}^{k}_{p}}=id_{\mathcal {E}^{k}_{p}}$ since $\wedge ^{k+1}$ is a homomorphism of groups. Therefore, $id\arrowvert _{\mathcal {E}^{k}_{p}}\in \Gamma _{k}(p,p)$ for all $0\le k< n$.

Theorem 5.7

(Theorem 4.16 in [2]) Let $\Gamma \subset \text {PU}(1,n)$ be a non-elementary discrete subgroup and $p\in L(\Gamma )$. Then,

$$\begin{aligned} \Gamma (p,p)=\{ \varphi : p^{\perp }(p) \longrightarrow p^{\perp }(p) \hspace{1mm}| \hspace{1mm} \varphi \text{ is } \text{ a } \text{ Cartan } \text{ map } \} \end{aligned}$$

is a compact Lie group.

Corollary 5.8

Let $\Gamma \subset \text {PU}(1,n)$ be a non-elementary discrete subgroup and $p\in L(\Gamma )$. Then, $\Gamma _{k}(p,p)$ is a group.

The following theorem extends the result in [2, Theorem 0.1] to our case $\wedge ^{k+1}\Gamma$ a discrete subgroup of PU(R,S).

Proof of Theorem 1.1 From Proposition 2.3 and Corollary 4.10, we have

$$\begin{aligned} \Lambda _{Kul}(\wedge ^{k+1}\Gamma )\subset \bigcup _{[\langle H^{k}_{p}\rangle ] \subset L_{k}(\Gamma )} \left[ \langle H^{k}_{p}\rangle \right] ^{\perp }. \end{aligned}$$

Now, we prove the other inclusion separating by cases depending on the cardinality of $L(\Gamma )$.

Case 1: $L(\Gamma )=\{p\}$ a single point. From Proposition 5.3 and arguments as in [2, Theorem 4.11], we can ensure that $Id_{p^{\perp }(p)} \in \Gamma _{0}(p,p)$. We prove $[\langle H^{k}_{p}\rangle ]^{\perp }\subset \Lambda _{Kul}(\wedge ^{k+1}\Gamma )$ by contradiction. Suppose there is a point $w_{0} \in [\langle H^{k}_{p}\rangle ]^{\perp }- [\langle H^{k}_{p}\rangle ]$ such that $w_{0} \in \Omega _{Kul}(\wedge ^{k+1}\Gamma )$. Let $(\gamma _{m})\subset \Gamma$ be a sequence whose Cartan map is $Id_{p^{\perp }(p)}$. So, with arguments as in Corollary 5.6, we have $Id_{\mathcal {E}^{k}_{p}}$ is the k-Cartan map associated with $(\gamma _{m})$. Now, by Proposition 5.3$[Span\{ \langle H^{k}_{p}\rangle , w_{0}\}]=\mathcal {D}_{(\wedge ^{k+1}\gamma _{m})}(w_{0})\subset \Lambda _{Kul} (\wedge ^{k+1}\Gamma )$, which is a contradiction to $w_{0} \in \Omega _{Kul}(\wedge ^{k+1}\Gamma )$. Hence,

$$\begin{aligned}{}[\langle H^{k}_{p}\rangle ] ^{\perp }\subset \Lambda _{Kul} (\wedge ^{k+1}\Gamma ). \end{aligned}$$

Case 2: If $|L(\Gamma )|=2$. After conjugating with an element in $\text {PU}(1,n)$, if is necessary, we can assume that $L(\Gamma )=\{[e_{1}],[e_{n+1}]\}$. Let $\Gamma _{0}=Isot(\Gamma ,[e_{1}])\cap Isot(\Gamma ,[e_{n+1}])$; thus, $\Gamma _{0}\subset \Gamma$ is a subgroup of finite index. Therefore, $\wedge ^{k+1}\Gamma _{0}\subset \wedge ^{k+1}\Gamma$ is also a subgroup of finite index, where

$$\begin{aligned} \wedge ^{k+1}\Gamma _{0}=Isot\left( \wedge ^{k+1}\Gamma ,[\langle H^{k}_{e_{1}}\rangle ]\right) \cap Isot(\wedge ^{k+1}\Gamma ,[\langle H^{k}_{e_{n+1}}\rangle ]). \end{aligned}$$

So, $\Lambda _{Kul}(\wedge ^{k+1}\Gamma _{0})=\Lambda _{Kul}(\wedge ^{k+1}\Gamma )$. Let $(\gamma _{m})\subset \Gamma _{0}$ be a sequence of distinct elements; then

$$\begin{aligned} \gamma _{m}=\left[ \begin{array}{ccc} r_{m}&{} &{} \\ &{} u_{m} &{} \\ &{} &{} r^{-1}_{m} \end{array} \right] ,\end{aligned}$$

(5.9)

where $r_{m}\in \mathbb {R}^{+}$, $|r_{m}|\ne 1$ and $u_{m}\in$ U$(n-1)$. From equation (5.9), we have $\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }[e_{1}]$, $\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ [e_{n+1}]$ uniformly on compact sets of $\mathbb{C}\mathbb{P}^{n} - [e_{n+1}^{\perp }]$ and $\mathbb{C}\mathbb{P}^{n} - [e_{1}^{\perp }]$ respectively.

Now, $(\wedge ^{k+1}\gamma _{m})\subset \wedge ^{k+1}\Gamma _{0}$ is a sequence of distinct elements. From Theorem 3.2, we get $\wedge ^{k+1}\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }[ \langle H^{k}_{e_{1}}\rangle ]$, $\wedge ^{k+1}\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ [ \langle H^{k}_{e_{n+1}}\rangle ]$ uniformly on compact sets of $\mathbb{C}\mathbb{P}^{N-1} - [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }$ and $\mathbb{C}\mathbb{P}^{N-1} - [ \langle H^{k}_{e_{1}}\rangle ]^{\perp }$ respectively.

The first thing we notice is that $[\langle H^{k}_{e_{1}}\rangle ]$, $[\langle H^{k}_{e_{n+1}}\rangle ]$, and $[\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cap [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }$ are sets $\wedge ^{k+1}\Gamma _{0}$-invariant. By the convergence above, $[\langle H^{k}_{e_{1}}\rangle ]$ is an attracting set, $[\langle H^{k}_{e_{n+1}}\rangle ]$ is a repelling set and, $[\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cap [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }$ is neither an attracting set nor repelling set of $(\wedge ^{k+1}\gamma _{m})$. That is,

• For each $z\in [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp } - \left\{ [\langle H^{k}_{e_{n+1}}\rangle ] \cup \left( [\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cap [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }\right) \right\}$ we have

  $$\begin{aligned} \wedge ^{k+1}\gamma _{m}(z) \underset{m\rightarrow \infty }{\longrightarrow }w , \hspace{2mm} \text{ with } \hspace{2mm} w \in [\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cap [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }. \end{aligned}$$

• For each $z\in [\langle H^{k}_{e_{1}}\rangle ]^{\perp } - \left\{ [\langle H^{k}_{e_{1}}\rangle ] \cup \left( [\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cap [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }\right) \right\}$ we have

  $$\begin{aligned} \wedge ^{k+1}\gamma _{m}(z) \underset{m\rightarrow \infty }{\longrightarrow }w , \hspace{2mm} \text{ with } \hspace{2mm} w \in [\langle H^{k}_{e_{1}}\rangle ]. \end{aligned}$$

  So, $[\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cap [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }$ contains the accumulation points of orbits of points in $[\langle H^{k}_{e_{n+1}}\rangle ]^{\perp } - [\langle H^{k}_{e_{n+1}}\rangle ]$.

From the previous dynamics, we have

$$\begin{aligned} \mathcal {L}_{0}(\wedge ^{k+1}\Gamma _{0})\cup \mathcal {L}_{1}(\wedge ^{k+1}\Gamma _{0})\subset [\langle H^{k}_{e_{1}}\rangle ] \cup [\langle H^{k}_{e_{n+1}}\rangle ] \cup \left( [\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cap [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }\right) , \end{aligned}$$

and $\mathcal {L}_{2}(\wedge ^{k+1}\Gamma _{0})=[\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cup [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }$. Hence,

$$\begin{aligned} \begin{aligned} \Lambda _{Kul}(\wedge ^{k+1}\Gamma )&= \Lambda _{Kul}(\wedge ^{k+1}\Gamma _{0})\\&=[\langle H^{k}_{e_{1}}\rangle ]^{\perp }\cup [\langle H^{k}_{e_{n+1}}\rangle ]^{\perp }. \end{aligned} \end{aligned}$$

We notice that the dynamics of $(\wedge ^{k+1}\gamma _{m})$ is very similar to the dynamics of a loxodromic element in PU(1,n); see [4] or [12].

Case 3: If $|L(\Gamma )|>2$, then $\Gamma$ is non-elementary. Let us now prove the other inclusion by contradiction. Let us suppose that

$$\begin{aligned} \bigcup _{[\langle H^{k}_{p}\rangle ] \subset L_{k}(\Gamma )} [\langle H^{k}_{p}\rangle ] ^{\perp } \not \subset \Lambda _{Kul}(\wedge ^{k+1}\Gamma ), \end{aligned}$$

so there is $p\in L(\Gamma )$ such that $[\langle H^{k}_{p}\rangle ]^{\perp } \not \subset \Lambda _{Kul}(\wedge ^{k+1}\Gamma )$. Then, there is a point $w_{0} \in [\langle H^{k}_{p}\rangle ]^{\perp } - [\langle H^{k}_{p}\rangle ]$ such that $w_{0} \in \Omega _{Kul}(\wedge ^{k+1}\Gamma )$. From Corollary 5.6, $Id_{\mathcal {E}^{k}_{p}} \in \Gamma _{k}(p,p)$; then, there is a sequence $(\gamma _{m})\subset \Gamma$ whose k-Cartan map is $Id_{\mathcal {E}^{k}_{p}}$. Now, from Proposition 5.3, we have $[Span\{\langle H^{k}_{p}\rangle , w_{0}\}] = \mathcal {D}_{(\wedge ^{k+1}\gamma _{m})}(w_{0}) \subset \Lambda _{Kul}(\wedge ^{k+1}\Gamma )$, which is a contradiction to $w_{0} \in \Omega _{Kul}(\wedge ^{k+1}\Gamma )$. Hence,

$$\begin{aligned} \bigcup _{[\langle H^{k}_{p}\rangle ] \subset L_{k}(\Gamma )} [\langle H^{k}_{p}\rangle ]^{\perp } \subset \Lambda _{Kul}(\wedge ^{k+1}\Gamma ). \end{aligned}$$

Therefore,

$$\begin{aligned} \Lambda _{Kul}(\wedge ^{k+1}\Gamma ) =\bigcup _{[\langle H^{k}_{p}\rangle ] \subset L_{k}(\Gamma )} [\langle H^{k}_{p}\rangle ] ^{\perp }. \end{aligned}$$

Now, from the above and the Corollary 4.10, we conclude that

$$\begin{aligned} \Omega _{Kul}(\wedge ^{k+1}\Gamma )=Eq(\wedge ^{k+1}\Gamma ). \end{aligned}$$

Through similar arguments, we can show that $\Omega _{Kul}(\wedge ^{k+1}\Gamma )=Eq(\wedge ^{k+1}\Gamma )$ is the largest open set on which $\wedge ^{k+1}\Gamma$ acts properly discontinuously.

6 Duality

In this section, we give a duality between the Kulkarni limit set and the k-Chen-Greenberg limit set.

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k< n$, then $\wedge ^{k+1}\Gamma \subset \text {PU}(R,S)$ where $R=\left( {\begin{array}{c}n\\ k\end{array}}\right)$ and $S=\left( {\begin{array}{c}n\\ k+1\end{array}}\right)$. Recall

$$\begin{aligned} L_{k}(\Gamma )=\overline{\bigcup _{\gamma \in Lim(\wedge ^{k+1}\Gamma )}Im(\gamma )}, \end{aligned}$$

is a collection of pairwise disjoint projective subspaces of dimension $M-1$, that is, $dim(Im(\gamma ))=M-1$, where $M=\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$. So, $L_{k}(\Gamma )\subset \mathbb {P}(\wedge ^{M}\mathbb {C}^{N})$ seen as points via the Plücker embedding, and $N=\left( {\begin{array}{c}n+1\\ k+1\end{array}}\right)$.

From the previous section,

$$\begin{aligned} \Lambda _{Kul}(\wedge ^{k+1}\Gamma )=\overline{\bigcup _{\gamma \in Lim(\wedge ^{k+1}\Gamma )}Ker(\gamma )}, \end{aligned}$$

where $dim(Ker(\gamma ))=N-M-1$, so $\Lambda _{Kul}(\wedge ^{k+1}\Gamma )\subset \mathbb {P}(\wedge ^{N-M}\mathbb {C}^{N})$ seen as points via the Plücker embedding.

To make notation simpler, we take $\widehat{\Gamma }=\wedge ^{k+1}\Gamma$, so $\wedge ^{N-M}\widehat{\Gamma }$ acts on the projective space $\mathbb {P}(\wedge ^{N-M}\mathbb {C}^{N})$.

Definition 6.1

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k < n$; we define the limit set of $\widehat{\Gamma }$ as

$$\begin{aligned} \widehat{L}_{k}(\widehat{\Gamma }):=\overline{\bigcup _{\gamma \in Lim(\wedge ^{N-M}\widehat{\Gamma })}Im(\gamma )}. \end{aligned}$$

For $F \subset \mathbb {P}(\wedge ^{N-M}\mathbb {C}^{N})\cap Q_{N-M-1,N-1}$ a subset, we denote by $V_{\ell }\subset \mathbb{C}\mathbb{P}^{N-1}$ the complex projective subspace that determines the point $\ell \in F$, then

$$\begin{aligned} \bigcup _{\ell \in F}V_{\ell } \end{aligned}$$

is the subset of $\mathbb{C}\mathbb{P}^{N-1}$ obtained as the union of all $(N-M-1)$-planes determined by the elements of F.

We have the following theorem:

Theorem 6.1

Let $\Gamma \subset \text {PU}(1,n)$ be a discrete subgroup, $0\le k<n$. Then,

$$\begin{aligned} \Lambda _{Kul}(\wedge ^{k+1}\Gamma )=\bigcup _{\ell \in \widehat{L}_{k}(\widehat{\Gamma })}V_{\ell }. \end{aligned}$$

Proof

Let $(\gamma _{m})\subset \Gamma$ be a sequence tending simply to infinity. By the Cartan decomposition theorem, there are sequences $(g^{}_{m}), (h^{}_{m}) \in K=\text {U}(1,n)\cap \text {U}(1+n)$ such that

$$\begin{aligned} \gamma _{m}=\left[ g^{}_{m}\mu (\gamma _{m})h^{}_{m}\right] \hspace{3mm} \text{ and } \hspace{3mm} \gamma _{m}^{-1}=\left[ (h_{m}^{})^{-1}(\mu (\gamma _{m}))^{-1}(g_{m}^{})^{-1}\right] . \end{aligned}$$

We can assume that

$$\begin{aligned} g_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ g \in K \hspace{2mm}\text{ and } \hspace{2mm} h_{m} \underset{m\rightarrow \infty }{\longrightarrow }\ h\in K. \end{aligned}$$

From Theorem 3.2, there exist two pseudo-projective transformations $[\rho ], [\delta ] \in QP(N,\mathbb {C})$ such that

$$\begin{aligned} \wedge ^{k+1}\gamma _{m} \underset{m\rightarrow \infty }{\longrightarrow }[\rho ] \hspace{3mm} \text{ and } \hspace{3mm} \wedge ^{k+1}\gamma ^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }[\delta ] \end{aligned}$$

in the sense of pseudo-projective transformations, where

$$\begin{aligned} \begin{aligned} Im(\rho )&= \wedge ^{k+1} g_{} \left( Span\left\{ e_{I} \right\} : I\in A \right) ,\\ Im(\delta )&=\wedge ^{k+1} h_{}^{-1} \left( Span\left\{ e_{I} \right\} : I\in B \right) ,\\ Ker(\rho )&=\wedge ^{k+1} h_{}^{-1} \left( Span\left\{ e_{I} \right\} : I \not \in A \right) ,\\ Ker(\delta )&=\wedge ^{k+1} g_{} \left( Span\left\{ e_{I} \right\} : I \not \in B \right) \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} A&=\{I=(1,i_{1},...,i_{k}) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne n+1\},\\ B&=\{I=(i_{1},...,i_{k},n+1) \hspace{3mm}\text{ with } \hspace{3mm} i_{j}\ne 1\}. \end{aligned} \end{aligned}$$

Let us take $\widehat{f}=\wedge ^{k+1}f$ for all $f\in \text {PU}(1,n)$. Now, from Theorem 3.3 in [16] applied to $(\wedge ^{N-M}\widehat{\gamma }_{m})\subset \wedge ^{N-M}\widehat{\Gamma }$, there exist two pseudo-projective transformations $[\alpha ], [\beta ]\in QP\big (\left( {\begin{array}{c}N\\ N-M\end{array}}\right) ,\mathbb {C}\big )$ such that

$$\begin{aligned} \wedge ^{N-M}\widehat{\gamma }_{m} \underset{m\rightarrow \infty }{\longrightarrow }[\alpha ] \hspace{3mm} \text{ and } \hspace{3mm} \wedge ^{N-M}\widehat{\gamma }^{-1}_{m} \underset{m\rightarrow \infty }{\longrightarrow }[\beta ] \end{aligned}$$

in the sense of pseudo-projective transformations, where

$$\begin{aligned} \begin{aligned} Im(\alpha )&= \wedge ^{N-M}\widehat{g}_{} \left( Span\left\{ e_{J} \right\} : J\in \widehat{A} \right) ,\\ Im(\beta )&=\wedge ^{N-M}\widehat{h}_{}^{-1} \left( Span\left\{ e_{J} \right\} : J\in \widehat{B} \right) ,\\ Ker(\alpha )&=\wedge ^{N-M}\widehat{h}_{}^{-1} \left( Span\left\{ e_{J} \right\} : J \not \in \widehat{A} \right) ,\\ Ker(\beta )&=\wedge ^{N-M} \widehat{g}_{} \left( Span\left\{ e_{J} \right\} : J \not \in \widehat{B} \right) \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \widehat{A}&=\{J=(I_{i_{1}},...,I_{i_{N-M}}) \hspace{3mm}\text{ with } \hspace{3mm} I_{i_{j}} \notin B \},\\ \widehat{B}&=\{J=(I_{i_{1}},...,I_{i_{N-M}}) \hspace{3mm}\text{ with } \hspace{3mm} I_{i_{j}} \notin A \}. \end{aligned} \end{aligned}$$

Now, we describe the image:

$$\begin{aligned} \begin{aligned} Im(\alpha )&= \wedge ^{N-M}\widehat{g}_{} \left( Span\left\{ e_{J} \right\} : J\in \widehat{A} \right) \\&= \wedge ^{N-M}\widehat{g}_{} \left( Span\left\{ e_{I_{i_{1}}}\wedge ...\wedge e_{I_{i_{N-M}}} \right\} : I_{i_{j}} \notin B \right) \\&= \widehat{g}_{}(e_{I_{i_{1}}})\wedge ...\wedge \widehat{g}_{}(e_{I_{i_{N-M}}}). \end{aligned} \end{aligned}$$

We notice that $[Im(\alpha )]= [\widehat{g}_{}(e_{I_{i_{1}}})\wedge ...\wedge \widehat{g}_{}(e_{I_{i_{N-M}}})]$ a point. So,

$$\begin{aligned} V_{[Im(\alpha )]}= [\widehat{g}_{}\left( Span\left\{ e_{I} \right\} : I \notin B \right) ]=[Ker(\delta )] \end{aligned}$$

seen as complex projective subspace of $\mathbb{C}\mathbb{P}^{N-1}$. Analogously,

$$\begin{aligned} V_{[Im(\beta )]}= [\widehat{h}_{}^{-1}\left( Span\left\{ e_{I}\right\} : I \notin A \right) ]=[Ker(\rho )] \end{aligned}$$

seen as complex projective subspace of $\mathbb{C}\mathbb{P}^{N-1}$.

On the other hand, from Proposition 4.3 and Theorem 1.1, we conclude that

$$\begin{aligned} \begin{aligned} \bigcup _{\ell \in \widehat{L}_{k}(\widehat{\Gamma })}V_{\ell }&=\overline{\bigcup _{\gamma \in Lim(\wedge ^{N-M}\widehat{\Gamma })} V_{Im(\gamma )}}\\&=\overline{\bigcup _{\rho \in Lim(\wedge ^{k+1}\Gamma )}Ker(\rho )}\\&=\Lambda _{Kul}(\wedge ^{k+1}\Gamma ). \end{aligned} \end{aligned}$$