Abstract
We study the distribution of non-discrete orbits of geometrically finite groups in \({\text {SO}}(n,1)\) acting on \({\mathbb {R}}^{n+1}\), and more generally on the quotient of \({\text {SO}}(n,1)\) by a horospherical subgroup. Using equidistribution of horospherical flows, we obtain both asymptotics for the distribution of orbits for the action of general geometrically finite groups, and we obtain quantitative statements with additional assumptions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We often seek to understand a group through the distribution of its orbits on a given space. In this paper, we will consider the action of certain geometrically finite groups on \({\mathbb {R}}^{n+1}\) and other spaces.
When \(\Gamma \) is a lattice in \({\text {SL}}_2({\mathbb {R}})\) acting on \({\mathbb {R}}^2\), this question was considered by Ledrappier [10], who proved that
for compactly supported functions f and \(X \in {\mathbb {R}}^2\), where \(c(\Gamma )\) is some constant depending on the covolume of the lattice \(\Gamma \), and \(\Vert \gamma \Vert \) denotes the \(\ell _2\) norm on \(\Gamma \). Nogueira [15] independently obtained this result for \(\Gamma ={\text {SL}}_2({\mathbb {Z}})\) using different methods. More recently, Macourant and Weiss obtained a quantitative version of this theorem for cocompact lattices in \({\text {SL}}_2({\mathbb {R}})\), and also for \(\Gamma ={\text {SL}}_2({\mathbb {Z}})\) in [13]. The case of lattices in \({\text {SL}}_n({\mathbb {R}})\) acting on different spaces V has also been considered, see for instance [5, 7].
In [16], Pollicott proved a similar quantitative theorem for the action of a lattice in \({\text {SL}}_2({\mathbb {C}})\) on \({\mathbb {C}}^2\). In the p-adic case, Ledrappier and Pollicott [11] considered lattices in \({\text {SL}}_2({\mathbb {Q}}_p)\) acting on \({\mathbb {Q}}_p^2\).
Similar questions have been studied extensively for lattices in a wide variety of groups G. For instance, Gorodnik and Weiss consider in [8] second countable, locally compact groups G with a general axiomatic approach, with several examples. More recently, Gorodnik and Nevo comprehensively studied the action of a lattice in a connected algebraic Lie group acting on infinite volume homogeneous varieties in [6], including obtaining quantitative results under appropriate assumptions.
The case when \(\Gamma \) has infinite covolume was recently studied by Maucourant and Schapira in [12], where they obtained an asymptotic version of Ledrappier’s result for convex cocompact subgroups of \({\text {SL}}_2({\mathbb {R}})\), with a scaling factor permitted. Moreover, they prove that an ergodic theorem like Ledrappier’s in the lattice case cannot be obtained in the infinite volume setting, because there is not even a ratio ergodic theorem. More specifically, [12, Proposition 1.5] shows that if \(\Gamma \subseteq {\text {SL}}_2({\mathbb {R}})\) is geometrically finite with \(-I\) the unique torsion element, then there exist small bump functions f and g such that for \(\overline{\nu }\)-almost every v (where \(\overline{\nu }\) is defined in Sect. 7),
does not have a limit. Thus, it is impossible to obtain an ergodic theorem in this setting with a normalization factor that does not depend on the functions. The key obstruction is the fluctuating behaviour of the Patterson–Sullivan measure. However, they show that with an additional averaging to address these fluctuations, there is a Log-Cesaro convergence, see [12, Theorem 1.6].
Throughout this paper, let \(G= {\text {SO}}(n,1)^\circ \) and let \(\Gamma \subseteq G\) be a Zariski dense geometrically finite subgroup. As a consequence of a more general ratio theorem we will discuss later in this section, we will obtain the following asymptotic behaviour for \(\Gamma \) orbits acting on
which is similar to a result of Maucourant and Schapira for \(n=2\). Note that V consists of null vectors of a certain quadratic form and corresponds to the upper half of the “light cone” in the usual representation of \({\text {SO}}(n,1)\); see Sect. 7.1 for more details.
When \(\Gamma \) is geometrically finite, the limit set of \(\Gamma \), denoted \(\Lambda (\Gamma ) \subseteq \partial ({\mathbb {H}}^n),\) decomposes into radial and bounded parabolic limit points:
For the precise definitions, see Sect. 2.
Proposition 1.1
Let \(\Gamma \) be convex cocompact. For any \(\overline{\varphi }\in C_c(V)\) and every \(v \in V\) with \(v^- \in \Lambda (\Gamma ),\) as \(T\rightarrow \infty \), we have that
where the implied constant depends on v and \(\Gamma \). Here, \(\delta _\Gamma \) denotes the critical exponent of \(\Gamma \), \(\Vert u\Vert _2\) denotes the Euclidean norm of \(u \in {\mathbb {R}}^{n+1},\) and \(\Gamma _T = \{\gamma \in \Gamma : \Vert \gamma \Vert \le T\}\), where \(\Vert \gamma \Vert \) denotes the max norm of \(\gamma \) as a matrix in \({\text {SL}}_{n+1}({\mathbb {R}})\). The notation \(v^- \in \Lambda _r(\Gamma )\) is discussed in Sect. 7.
Here, the notation \(a \asymp b\) means that there exists a constant \(\lambda >1\) such that
The precise definition of the measure \(\overline{\nu }\) is discussed in Sect. 7. It is the pushforward of the measure \(\nu \) defined in Sect. 2.3, which is part of the product structure of the Burger–Roblin (BR) measure, defined fully in that section.
Let \(U=\{u_\mathbf{t }: \mathbf{t }\in {\mathbb {R}}^{n-1}\}\) be the expanding horospherical subgroup for the frame flow A. Let \(P\subset G\) be the parabolic subgroup which contains the contracting horospherical subgroup. Parametrizations of these groups are given in Sect. 2.
Proposition 1.1 is obtained by counting orbit points in \(U\backslash G\). We will also establish a stronger version, specifically showing that a more precise ratio tends to 1. With additional assumptions on \(\Gamma \), we obtain a quantitative version of this statement. We need to define additional notation in order to state this result.
Let UAK be the Iwasawa decomposition of \({\text {SL}}_{n+1}({\mathbb {R}})\), and let \(\Psi : U \backslash G \rightarrow G\) be the map
where \(g=uak\) in the Iwasawa decomposition.
We view G as embedded in \({\text {SL}}_{n+1}({\mathbb {R}})\). For \(g \in G\), let \(\Vert g\Vert \) denote the max norm as a matrix in \({\text {SL}}_{n+1}({\mathbb {R}})\). The following “product” is useful for our statements (a similar definition exists in the \({\text {SL}}_2({\mathbb {R}})\) case). For \(x,y\in U\backslash G\), let
where \(E_{1,n+1}\) is the \((n+1)\times (n+1)\) matrix with one in the \((1,n+1)\)-entry and zeros everywhere else. For \(x \in U\backslash G\) and \(g \in G\), \(x \star xg\) measures the difference between the U components of the Iwasawa decomposition of x and xg. More specifically, it measures the \((1,n+1)\) component of g.
For \(L \subseteq G\), define
and
where \(\Vert \mathbf{t }\Vert \) denotes the max norm of \(\mathbf{t }\in {\mathbb {R}}^{n-1}.\) Let \(\pi _U : G \rightarrow U\backslash G\) denote the natural projection map.
We will be interested in the following quantity:
Here, \(\varphi \) is a function on \(U\backslash G\), \(x\in U\backslash G\), \(T > 0\), \(\mu ^{{\text {PS}}}\) denotes the PS measure, fully defined in Sect. 2.2, and \(\nu \) is defined in Sect. 2.3.
For two functions of T, a(T), b(T), we write
We can now state a qualitative version of our ratio theorem:
Theorem 1.2
Let \(\Gamma \) be geometrically finite. For any \(\varphi \in C_c(U\backslash G)\) and every \(x \in U\backslash G\) such that \(\Psi (x)^- \in \Lambda _r(\Gamma ),\)
The notation \(g^-\) for \(g \in G\) is defined in Sect. 2.
By the shadow lemma, Proposition 2.1, we obtain the following corollary, which will in turn imply Proposition 1.1:
Corollary 1.3
Assume that \(\Gamma \) is convex cocompact. For any \(\varphi \in C_c(U\backslash G)\) and every \(x \in U\backslash G\) such that \(\Psi (x)^-\in \Lambda (\Gamma ),\) as \(T \rightarrow \infty \),
where the implied constant depends on x and \(\Gamma \).
Remark 1.4
The proof also works for \(\Gamma \) geometrically finite when the geodesic of \(\Psi (x)\Gamma \) is bounded. We must then assume that \(\Psi (x)^- \in \Lambda _r(\Gamma ).\)
In order to state the quantitative version of Theorem 1.2, we need an additional definition, which gives a precise formulation of the notion that \(x \in G/\Gamma \) does not escape to the cusps “too quickly”:
Definition 1.5
For \(0<\varepsilon <1\) and \(s_0\ge 1\), we say that \(x\in G/\Gamma \) with \(x^- \in \Lambda (\Gamma )\) is \((\varepsilon ,s_0)\)-Diophantine if for all \(s>s_0\),
where \({\mathcal {C}}_0\) is a compact set arising from the thick-thin decomposition, and is fully defined in Sect. 2.1. We say that \(x\in G/\Gamma \) is \(\varepsilon \)-Diophantine if it is \((\varepsilon ,s_0)\)-Diophantine for some \(s_0\).
Remark 1.6
A point \(x\in G/\Gamma \) is \(\varepsilon \)-Diophantine for some \(\varepsilon >0\) if and only if \(x^-\in \Lambda _r(\Gamma )\), because Definition 1.5 precisely says that \(x^- \not \in \Lambda _{bp}(\Gamma )\), by the construction of the thick-thin decomposition.
When \(\Gamma \) is convex cocompact, every \(x \in G/\Gamma \) with \(x^- \in \Lambda (\Gamma )\) is \(\varepsilon \)-Diophantine for some \(\varepsilon \), because all limit points are radial in this case. Observe also that in the lattice case, this condition is always satisfied, because \(\Lambda (\Gamma ) = \partial ({\mathbb {H}}^n)\). See [18] for further discussion of this definition.
Definition 1.7
We say that \(\Gamma \) satisfies property A if one of the following holds:
-
\(\Gamma \) is convex cocompact, or
-
\(\Gamma \) is geometrically finite, and either
-
1.
\(n \le 4\) and \({\mathbb {H}}^n/\Gamma \) has a cusp of rank \(n-1\), or
-
2.
\(\delta _{\Gamma }>n-2\).
-
1.
Remark 1.8
The assumptions on \(\Gamma \) in Definition 1.7 are to ensure the effective equidistribution theorem in [18, Theorem 1.4] holds (see Theorem 2.15 for a statement of this theorem in this setting). As discussed in [18], this theorem holds whenever the frame flow satisfies an explicit exponential mixing statement, [18, Assumption 1.1], and this condition is satisfied under the conditions in Definition 1.7. However, Definition 1.7 could be replaced with assuming that the more technical statement [18, Assumption 1.1] is satisfied.
Throughout the paper, the notation
means there exists a constant c such that
If a subscript is denoted, e.g. \(\ll _\Gamma \), this explicitly indicates that this constant depends on \(\Gamma \).
Theorem 1.9
Let \(\Gamma \) satisfy property A. For any \(0<\varepsilon <1\), there exist \(\ell =\ell (\Gamma )\in {\mathbb {N}}\) and \(\kappa =\kappa (\Gamma ,\varepsilon )\) satisfying: for every \(\varphi \in C^\infty _c(U\backslash G)\) and for every \(x\in U\backslash G\) such that \(\Psi (x)\Gamma \) is \(\varepsilon \)-Diophantine, and for all \(T\gg _{\Gamma ,{\text {supp}}{\varphi },x}1\),
The dependencies in this statement are quite explicit. The dependence of T on x in Theorem 1.9 arises from the constant in Lemma 3.2, which is explicitly defined in that proof, and the precise Diophantine nature of x, through Theorem 2.15 (i.e. the \(\varepsilon \) and \(s_0\) that appear in Definition 1.5). The implied dependence on x in the conclusion is discussed at the end of Sect. 6.
If the support of the function is small enough, then we can get a more explicit estimate (see Sect. 5). This is used as a main step in the proof of Theorem 1.9.
This paper is organized as follows. In Sect. 2, we present notation used throughout the paper, the definitions and fundamental properties of the measures we are working with, and the equidistribution theorems that will be key in our arguments. In Sect. 3, we explore the duality between \(\Gamma \) orbits on \(U\backslash G\) and of U orbits on \(G/\Gamma \), and prove key lemmas that are common to the proofs of both Theorems 1.2 and 1.9 . This involves a thickening argument, due to Ledrappier, to reduce the problem to that of equidistribution of U orbits. In Sect. 4, we prove Theorem 1.2, using an equidistribution theorem of Mohammadi and Oh, Theorem 2.14. In Sect. 6, we prove Theorem 1.9, using a quantitative equidistribution theorem, Theorem 2.15. Finally, in Sect. 7, we consider two specific examples, and prove Proposition 1.1.
2 Notation and preliminary results in \(G/\Gamma \)
Let \(G= {\text {SO}}(n,1)^\circ \) and let \(\Gamma \subseteq G\) be a Zariski dense discrete subgroup. Let \(\pi _\Gamma :G\rightarrow G/\Gamma \) be the quotient map.
Let \(\Lambda (\Gamma )\subseteq \partial ({\mathbb {H}}^n)\) denote the limit set of \(G/\Gamma \), i.e., the set of all accumulation points of \(\Gamma z\) for some \(z\in {\mathbb {H}}^n\cup \partial ({\mathbb {H}}^n)\).
The convex core of \(X:=G/\Gamma \) is the image in X of the minimal convex subset of \({\mathbb {H}}^n\) which contains all geodesics connecting any two points in \(\Lambda (\Gamma )\).
We say that \(\Gamma \) is geometrically finite if a unit neighborhood of the convex core of \(\Gamma \) has finite volume.
Fix a reference point \(o \in {\mathbb {H}}^n\). Let \(K=\text {Stab}_G(o)\) and let d denote the left G-invariant metric on G which induces the hyperbolic metric on \(K\backslash G={\mathbb {H}}^n\). Fix \(w_o \in {\text {T}}^1({\mathbb {H}}^n)\) and let \(M = {\text {Stab}}_G(w_o)\) so that \({\text {T}}^1({\mathbb {H}}^n)\) may be identified with \(M\backslash G\). For \(w \in {\text {T}}^1({\mathbb {H}}^n)\),
denotes the forward and backward endpoints of the geodesic w determines. For \(g \in G\), we define
We say that a limit point \(\xi \in \Lambda (\Gamma )\) is radial if there exists a compact subset of X so that some (and hence every) geodesic ray toward \(\xi \) has accumulation points in that set. We denote by \(\Lambda _r (\Gamma )\) the set of all radial limit points.
An element \(g\in G\) is called parabolic if the set of fixed points of g in \(\partial ({\mathbb {H}}^n)\) is a singleton. We say that a limit point is parabolic if it is fixed by a parabolic element of \(\Gamma \). A parabolic limit point \(\xi \in \Lambda (\Gamma )\) is called bounded if the stabilizer \(\Gamma _\xi \) acts cocompactly on \(\Lambda (\Gamma )-{\xi }\).
We denote by \(\Lambda _r (\Gamma )\) and \(\Lambda _{bp} (\Gamma )\) the set of all radial limit points and the set of all bounded parabolic limit points, respectively. Since \(\Gamma \) is geometrically finite (see [2]),
Let \(A=\left\{ a_s\,:\,s\in {\mathbb {R}}\right\} \) be a one parameter diagonalizable subgroup such that M and A commute, and such that the right \(a_s\) action on \(M\backslash G={\text {T}}^1({\mathbb {H}}^n)\) corresponds to unit speed geodesic flow.
We embed G in \({\text {SL}}_{n+1}({\mathbb {R}})\), parametrize A by \(A=\{a_s : s\in {\mathbb {R}}\}\), where
and I denotes the \((n-1)\times (n-1)\) identity matrix, and let
Let U denote the expanding horospherical subgroup
let \({{\tilde{U}}}\) be the contracting horospherical subgroup
and let \(P=MA{{\tilde{U}}}\) be the parabolic subgroup.
The group U is isomorphic to \({\mathbb {R}}^{n-1}\). We use the parametrization \(U=\{u_\mathbf{t }: \mathbf{t }\in {\mathbb {R}}^{n-1}\}\), where \(\mathbf{t }\) is viewed as a row vector, and
For more details on these parametrizations and the interactions between these groups, see [18, §2].
2.1 Thick-thin decomposition and the shadow lemma
There exists a finite set of \(\Gamma \)-representatives \(\xi _1,\dots ,\xi _q\in \Lambda _{bp}(\Gamma )\). For \(i=1,\dots ,q\), fix \(g_i\in G\) such that \(g_i^- =\xi _i\), and for any \(R>0\), set
(recall, \(K=\text {Stab}_G (o)\)). Each \({\mathcal {H}}_i(R)\) is a horoball of depth R.
The rank of \({\mathcal {H}}_i(R)\) is the rank of the finitely generated abelian subgroup \(\Gamma _{\xi _i}={\text {Stab}}_\Gamma (\xi _i)\). It is known that each rank is strictly smaller than \(2\delta _\Gamma \).
Let
Note that the condition \(g^\pm \in \Lambda (\Gamma )\) is independent of the choice of representative of \(x=g\Gamma \) in the above definition, because \(\Lambda (\Gamma )\) is \(\Gamma \)-invariant. Thus, the notation \(x^\pm \in \Lambda (\Gamma )\) is well-defined. For now, \({\text {supp}}m^{{\text {BMS}}}\) is simply notation, but as we will see, this coincides with the support of the BMS measure, \(m^{{\text {BMS}}}.\) We say that a point \(x\in X\) is a BMS point if \(x\in {\text {supp}}m^{{\text {BMS}}}\).
According to [2], there exists \(R_0\ge 1\) such that \({\mathcal {X}}_1(R_0),\dots ,{\mathcal {X}}_q(R_0)\) are disjoint, and for some compact set \({\mathcal {C}}_0\subset X\),
2.2 Patterson–Sullivan measure
A family of finite measures \(\{\mu _x\,:\,x\in {\mathbb {H}}^n\}\) on \(\partial ({\mathbb {H}}^n)\) is called a \(\Gamma \)-invariant conformal density of dimension \(\delta _\mu >0\) if for every \(x,y\in {\mathbb {H}}^n\), \(\xi \in \partial ({\mathbb {H}}^n)\) and \(\gamma \in \Gamma \),
where \(\gamma _*\mu _x(F)=\mu _x(F\gamma )\) for any Borel subset F of \(\partial ({\mathbb {H}}^n)\).
We let \(\{\nu _x\}_{x \in {\mathbb {H}}^n}\) denote the Patterson–Sullivan density on \(\partial {\mathbb {H}}^n\), that is, the unique (up to scalar multiplication) conformal density of dimension \(\delta _\Gamma \).
For each \(x\in {\mathbb {H}}^n\), we denote by \(m_x\) the unique probability measure on \(\partial ({\mathbb {H}}^n)\) which is invariant under the compact subgroup \(\text {Stab}_G (x)\). Then \(\left\{ m_x\,:\,x\in {\mathbb {H}}^n\right\} \) forms a G-invariant conformal density of dimension \(n-1\), called the Lebesgue density. Fix \(o \in {\mathbb {H}}^n\).
For \(x, y \in {\mathbb {H}}^n\) and \(\xi \in \partial ({\mathbb {H}}^n)\), the Busemann function is given by
where \(\xi _t\) is a geodesic ray towards \(\xi \).
For \(g\in G\), we can define measures on Ug using the conformal densities defined previously. The Patterson–Sullivan measure (abbreviated as the PS-measure):
and the Lebesgue measure
Note that for any \(g\in G\), a point \(h\in Ug\) satisfies \(h\in {\text {supp}}\mu ^{{\text {PS}}}_{Ug}\) if and only if \(h^+\in \Lambda (\Gamma )\). Therefore, we refer to the points \(x\in X\) which satisfy \(x^+\in \Lambda (\Gamma )\) as PS points.
The conformal properties of \(m_x\) and \(\nu _x\) imply that this definition is independent of the choice of \(o\in {\mathbb {H}}^n\).
We often view \(\mu ^{{\text {PS}}}_{Ug}\) as a measure on U via
The measure
is independent of the orbit Ug and is simply the Lebesgue measure on \(U\equiv {\mathbb {R}}^{n-1}\) up to a scalar multiple.
If \(x \in X\) is such that \(x^- \in \Lambda _r(\Gamma )\), then
is injective, and we can define the PS measure on \(Ux \subseteq X\), denoted \(\mu ^{{\text {PS}}}_x\), simply by pushforward of \(\mu ^{{\text {PS}}}_g\), where \(x = g\Gamma \). In general, defining \(\mu ^{{\text {PS}}}_x\) requires more care, see e.g. [14, §2.3] for more details. As before, we can view \(\mu ^{{\text {PS}}}_x\) as a measure on U via
Recall that for \(T>0\),
where \(\Vert \mathbf{t }\Vert \) is the max norm of \(\mathbf{t }\) as measured in \({\mathbb {R}}^{n-1}\).
We will need the following version of Sullivan’s shadow lemma:
Proposition 2.1
([12, Proposition 5.1, Remark 5.2])
There exists a constant \(\lambda =\lambda (\Gamma )\ge 1\) such that for all \(x \in {\text {supp}}m^{{\text {BMS}}}\) and all \(T>0\), we have
where k(x, T) denotes the rank of the cusp containing \(a_{-\log T}x\) (and is zero if \(a_{-\log T}x \in {\mathcal {C}}_0\)).
Remark 2.2
In [12], the shadow lemma is proven using the distance measured in \({\mathbb {H}}^n/\Gamma .\) However, because \({\mathcal {C}}_0\) is K-invariant and \({\mathbb {H}}^n = K \backslash G\), we obtain the form above.
Remark 2.3
When \(\Gamma \) is convex cocompact, \({\mathcal {C}}_0={\text {supp}}m^{{\text {BMS}}}\), and the shadow lemma simplifies to
We will need the following, which says that the PS measure is doubling.
Lemma 2.4
([18, Corollary 9.9]) There exist constants \(\sigma _1 = \sigma _1(\Gamma )\ge \delta _\Gamma \), \(\sigma _2 = \sigma _2(\Gamma )>0\) such that for every \(c >0\), every \(x \in {\text {supp}}m^{{\text {BMS}}}\) and every \(T>0\),
We will also require control of the PS measure of slightly larger balls, specifically as will be established below in Lemma 2.9. This will be a result of the friendliness of the PS density when \(\Gamma \) is geometrically finite, established in [3]. More specifically, we will show that the measure of the boundary of certain balls can be controlled.
Let d be a left-invariant Riemannian metric on \(G/\Gamma \) that projects to the hyperbolic distance on \({\mathbb {H}}^n\).
Denote by \(d_E\) the Euclidean metric on \({\mathbb {R}}^{n-1}\). For a subset \(S\subseteq {\mathbb {R}}^{n-1}\) and \(\xi >0\), let
For \(v\in {\mathbb {R}}^{n-1}\) and \(r>0\), let
be the Euclidean ball of radius r around v.
We say that a hyperplane L is on the boundary of a closed ball B if
Below, we obtain estimates for the PS measure of small neighbourhoods of hyperplanes on the boundary of a ball centered at a BMS point. Though not written here, estimates also hold when the center of the ball is a PS point but not a BMS point, as long as the ball is sufficiently small. In this case, one may use arguments similar to those in the appendix of [18].
We caution the reader that the estimates below hold only for hyperplanes on the boundary of such a ball; to obtain such estimates for general hyperplanes, absolute friendliness of the PS density is necessary. By [3, Theorem 1.9], this is satisfied if and only if all cusps of \({\mathbb {H}}^n/\Gamma \) are of maximal rank \(n-1\) (note that this is vacuously satisfied if \(\Gamma \) is convex cocompact). In this case, one may use [18, Corollary 9.14] when \(\Gamma \) is geometrically finite or [17, Theorem 2] if \(\Gamma \) is convex cocompact.
Lemma 2.5
There exists a constant \(\alpha =\alpha (\Gamma )>0\) satisfying the following: for all \(\lambda \in \Lambda (\Gamma )\), \(\xi >0,\) \(0<\eta \le 1\), and every hyperplane L that is on the boundary of \(B(\lambda ,\eta )\), we have that
Proof
By [3, Theorem 1.9], \(\nu _o\) is friendly when \(\Gamma \) is geometrically finite. In particular, this means that there exists \(\alpha =\alpha (\Gamma )>0\) such that for all \(\lambda \in \Lambda (\Gamma )\), \(\xi >0\), \(0<\eta \le 1\), and every affine hyperplane \(L \subseteq \partial ({\mathbb {H}}^n)\),
where
Since \(\lambda \in \Lambda (\Gamma ),\) for any L that is on the boundary of \(B(\lambda ,\eta \)), we have that
Thus, for any L that is on the boundary of \(B(\lambda ,\eta )\), we have
Replacing \(\xi \) with \(2\xi \eta ^{-1}\) then implies that for every such L,
as desired. \(\square \)
By flowing with \(a_{-s}\) for \(s>0\), we obtain similar estimates for large balls centered at BMS points:
Corollary 2.6
Let \(\alpha =\alpha (\Gamma )>0\) be as in Lemma 2.5. For every \(x\in {\text {supp}}m^{{\text {BMS}}}\) such that \(x^-\in \Lambda _r(\Gamma )\), every \(\eta ,\xi >0,\) and every hyperplane L in the boundary of \(B_U(\eta )x,\) we have
Proof
We will first prove that there exists a constant \(c=c(\Gamma )>0\) so that for all \(x \in {\text {supp}}m^{{\text {BMS}}}\cap {\mathcal {C}}_0\) with \(x^- \in \Lambda _r(\Gamma ),\) \(\xi >0\), \(\eta \) satisfying
and every hyperplane L in the boundary of \(B_U(\eta )x\), the inequality in the statement is satisfied.
Since \({\mathcal {C}}_0\) is compact, there exists \(c=c(\Gamma )>1\) such that for any \(x\in {\mathcal {C}}_0\) we can find \(g\in G\) such that \(x=g\Gamma \) and \(d(o,g(o))<c\). Then, for any \(u_\mathbf{t }\in B_U(c^{-1})\), we have
Then, by the definition of the PS measure, we have
where \({\text {Pr}}_{g^-} : Ug \rightarrow \partial ({\mathbb {H}}^n)\setminus \{g^-\}\) is the visual map \(w \mapsto w^+\). Using [18, Corollary 9.5], we may assume that c satisfies
where \(L'\) is a hyperplane in the boundary obtained from the projection of L under the visual map \({\text {Pr}}_{g^-}\). Thus, for any \(x\in {\mathcal {C}}_0\) and \(u_\mathbf{t }\in B_U(c)\), we arrive at
and in a similar way, one may also deduce
Now, we may conclude
Now, let \(x \in {\text {supp}}m^{{\text {BMS}}}\) with \(x^- \in \Lambda _r(\Gamma )\) and let \(\eta >0\). Since \(a_{-s}x\) has accumulation points in \({\mathcal {C}}_0\), there exists \(s>0\) so that \(e^{-s}\eta <c^{-1}\) and \(a_{-s}x \in {\mathcal {C}}_0\). By the first step of the proof, we then have that
\(\square \)
Proposition 2.7
Let \(\alpha =\alpha (\Gamma )>0\) be as in Corollary 2.6. Then for all \(x \in {\text {supp}}m^{{\text {BMS}}}\) such that \(x^-\in \Lambda _r(\Gamma )\), \(T>0\), and \(0<\varepsilon \le 1\), we have that
Proof
By the geometry of \((B_U((1+2\varepsilon )T)-B_U(T))x\), there exists a constant m depending only on n and hyperplanes \(L_1,\ldots , L_m\) in the boundary of \(B_U((1+2\varepsilon )T)x\) so that
Then by Corollary 2.6, we have that
Note that the assumption \(\varepsilon \le 1\) is for convenience in the last step only; one may still use Lemma 2.4 if \(\varepsilon \) is not bounded, but the exponent on \(\varepsilon \) must change. \(\square \)
We can obtain estimates for all \((\varepsilon ,s_0)\)-Diophantine points for balls that are sufficiently large (in a way that is uniform and linear in \(s_0\)). In fact, for any compact set \(\Omega \subseteq G/\Gamma \), there exists a \(T_0=T_0(\Omega )\) satisfying the statement below for all \(x\in \Omega \) with \(x^- \in \Lambda _r(\Gamma ),\) see e.g. [14, Lemma 3.3]. Thus, the statement below could take many forms and this is not as strong as possible; we simply write it in a way that is useful for our setting.
Corollary 2.8
Let \(\alpha = \alpha (\Gamma )>0\) be as in Proposition 2.7, let \(0<\varepsilon \le 1\) and let \(s_0\ge 1.\) There exists \(T_0 = T_0(\Gamma ,s_0)>0\) so that for every \((\varepsilon ,s_0)\)-Diophantine point \(x \in G/\Gamma \), all \(T\ge 2T_0+1\), and all \(\xi >0\),
In particular, if \(x^- \in \Lambda _r(\Gamma ),\) there exists \(T_0=T_0(x)>0\) so that for all \(T \ge 2T_0+1\) and all \(\xi >0,\) (10) holds.
Proof
By [18, Lemma 3.8], there exists \(T_0=T_0(\Gamma ,s_0)>0\) (in fact, it is linear in \(s_0\)) so that for every \((\varepsilon ,s_0)\)-Diophantine point x, there exists
For \(T \ge T_0\), we have
In particular,
and
Since all Diophantine points are radial, by assuming that \(T \ge 2T_0+1\), we may use Proposition 2.7 below:
Since \(T \ge 2T_0+1,\)
and it can be absorbed into the \(\xi \) term, completing the proof. \(\square \)
We may now state the form in which we will need this control. The implied constant below depends on x through the initial time in Corollary 2.8, so it can be made uniform over a compact set or over all points with the same Diophantine properties. However, this level of detail is not necessary for our results.
Lemma 2.9
Let \(\alpha = \alpha (\Gamma )>0\) be as in Corollary 2.8. For every \(x \in G/\Gamma \) with \(x^- \in \Lambda _r(\Gamma ),\) \(c>0\), \(0<\eta \le 1,\) and \(0<r_+<\ell <r_-\) satisfying
there exists \(T_0=T_0(x,r_+,r_-)>0\) such that for any \(T>T_0\),
Proof
First, observe that by Corollary 2.8, there exists \(T_1=T_1(x)\) so that for all \(T \ge 2T_1+1\) and all \(\xi >0\),
This follows immediately from the fact that
Thus, if we assume that T is sufficiently large so that \(\sqrt{T}/\ell \ge 2T_1+1\) (and note that this condition can be taken to rely on \(r_-\) rather than on \(\ell \) specifically), and note that by the assumption,
we see from (12) that
Note that the implied constant depends on x because we have absorbed the constant \(T_1.\) Now, choose \(T_0 \ge T_1\) so that \(T \ge T_0\) implies \(\frac{\ell }{\sqrt{T}}<1\) (a condition which depends on x and \(r_-\) in this case), which implies the claim because we may then absorb this term into the \(\eta \) term.
The second case can be shown in a similar way, with the choice of \(T_0\) depending on x and \(r_+\) there. \(\square \)
2.3 Burger–Roblin and Bowen–Margulis–Sullivan measures
Let \(\pi :{\text {T}}^1({\mathbb {H}}^n)\rightarrow {\mathbb {H}}^n\) be the natural projection. Recalling the fixed reference point \(o \in {\mathbb {H}}^n\) as before, the map
is a homeomorphism between \({\text {T}}^1({\mathbb {H}}^n)\) and
This homeomorphism allows us to define the Bowen-Margulis-Sullivan (BMS) and the Burger–Roblin (BR) measure on \({\text {T}}^1({\mathbb {H}}^n)\), denoted by \({\tilde{m}}^{{\text {BMS}}}\) and \({\tilde{m}}^{{\text {BR}}}\), respectively:
The conformal properties of \(\left\{ \nu _x\right\} \) and \(\left\{ m_x\right\} \) imply that these definitions are independent of the choice of \(o\in {\mathbb {H}}^n\). Using the identification of \({\text {T}}^1({\mathbb {H}}^n)\) with \(M\backslash G\), we lift the above measures to G so that they are all invariant under M from the left. By abuse of notation, we use the same notation (\({\tilde{m}}^{{\text {BMS}}}\) and \({\tilde{m}}^{{\text {BR}}}\)). These measures are left \(\Gamma \)-invariant, and hence induce locally finite Borel measures on X, which are the Bowen-Margulis-Sullivan measure \(m^{{\text {BMS}}}\) and the Burger–Roblin measure \(m^{{\text {BR}}}\), respectively.
Note that
and the support of the BMS measure indeed satisfies (4).
Recall the definition of \({\tilde{U}}\), and \(P = MA{\tilde{U}}\) from the begining of Sect. 2. P is the stabilizer of \(w_o^+\) in G. Hence, one can define a measure \(\nu \) on Pg for \(g \in G\), which will give us a product structure for \({\tilde{m}}^{{\text {BMS}}}\) and \({\tilde{m}}^{{\text {BR}}}\) that will be useful in our approach. For any \(g\in G\), define
on Pg, where \(s = \beta _{(pg)^-}(o,pg(o))\), \(p = mav \in MA{\tilde{U}}\) and dm is the probability Haar measure on M.
Then for any \(\psi \in C_c(G)\) and \(g\in G\), we have
and
2.4 Sobolev norms
In the next section we formulate the equidistribution and effective equidistribution results which we will use in the proof of the main theorems. In order to formulate them, we first need to define Sobolev norms. Our proofs will require constructing smooth indicator functions and partitions of unity with controlled Sobolev norms. This section also includes lemmas constructing such partitions.
For \(\ell \in {\mathbb {N}}\), \(1\le p\le \infty \), and \(\psi \in C^\infty (X)\cap L^p(X)\) we consider the following Sobolev norm
where the sum is taken over all monomials U in a fixed basis of \({\mathfrak {g}}=\text{ Lie }(G)\) of order at most \(\ell \), and \(\left\| \cdot \right\| _p\) denotes the \(L^p(X)\)-norm. Since we will be using \(S_{2,\ell }\) most often, we set
Lemma 2.10
([9, Lemma 2.4.7])
-
(1)
Let X, Y be Riemannian manifolds, and let \(\varphi \in C_c^\infty (X)\), \(\psi \in C_c^\infty (Y)\). Consider \(\varphi \cdot \psi \) as a function on \(X \times Y\). Then
$$\begin{aligned}S_\ell (\varphi \cdot \psi )\le c(X,Y)S_\ell (\varphi )S_\ell (\psi ), \end{aligned}$$where c(X, Y) is a constant depending only on X and Y (independent of \(\varphi ,\psi \)).
-
(2)
Let X be a Riemannian manifold of dimension N and let \(x \in X\). Then for any \(0<r<1\), there exists a non-negative function \(f\in C_c^\infty (X)\) such that \({\text {supp}}(f)\) is contained in the ball of radius r centered at x, \(\int _X f = 1,\) and
$$\begin{aligned} S_\ell (f)\le c(X,x)r^{-\ell +N/2}, \end{aligned}$$where c(X, x) is a constant depending only on X and x, not r.
The following lemma is an immediate consequence of the product rule.
Lemma 2.11
Let X be a Riemannian manifold and let \(\varphi ,\psi \in C_c^\infty (X)\). For any \(\ell \in {\mathbb {N}}\),
Lemma 2.12
For any \(\ell '\) there exists \(\ell >\ell '\) which satisfies the following. Let X, Y be Riemannian manifolds, \(\varphi \in C_c^\infty (X)\), and \(\psi :Y\rightarrow X\) be a smooth function. Then
Proof
By the chain rule, for any \(1\le k\le \ell '\),
where in the last line, we have used [1] to choose \(\ell >\ell '\) satisfying
for any f, where the implied constant is global. \(\square \)
Lemma 2.13
Let H be a Riemannian manifold of dimension N, \(0<r<1\), \(\ell \in {\mathbb {N}}\), and E a bounded subset of H. Then, there exists a partition of unity \(\sigma _1,\dots ,\sigma _k\) of E in \(H_r(E)=\{g \in G : d_H(g,E)\le r\}\) where \(d_H\) denotes the Riemannian metric on H, i.e.
such that for some \(u_1,\dots ,u_k\in E\) and all \(1\le i\le k\)
Moreover,
Proof
According to Lemma 2.10(b) there exists a non-negative smooth function \(\sigma \) supported on \(H_{r/2}\) such that
Since H is a Riemannian manifold and E is bounded, there exists a smooth partition of unity, \(f_i:H\rightarrow {\mathbb {R}}\), \(i=1,\dots ,k\), such that each \(f_i\) is supported on a ball of radius r/2 with a center \(u_i\in E\) and
For \(i=1,\dots ,k\) define \(\sigma _i\) by
We will show that \(\sigma _1,\dots ,\sigma _k\) satisfy the claim.
By definition, for \(i=1,\dots ,k\), \(\sigma _i\) is supported on a ball of radius r and centered at a point in E. By Young’s convolution inequality, we have
For any \(h\in E\), \(h^{-1}E\) contains the identity, and so we have \(h^{-1}H_{r/2}(E)\supseteq H_{r/2}\). Thus,
If \(h\notin H_{r} (E)\), then we have \(h^{-1}H_{r/2}(E)\cap H_{r/2}=\emptyset \). Hence, the above computation yields
Note that by (16), and since \(f_i\) is a partition of unity, we may also deduce
\(\square \)
2.5 Equidistribution results
For the proof of the main theorems we use the equidistribution results stated below.
The following theorem was proved for \(G={\text {SL}}_2({\mathbb {R}})\) by Maucourant and Schapira in [12] and for \(G={\text {SO}}(n,1)^\circ \) by Mohammadi and Oh in [14].
Theorem 2.14
Let \(\Gamma \) be geometrically finite. Fix \(x\in G/\Gamma \) such that \(x^-\in \Lambda _r(\Gamma )\). Then for any \(\psi \in C_c(G/\Gamma )\) we have
Theorem 2.15
([18, Theorem 1.4 and Remark 7.3]) Assume \(\Gamma \) satisfies property A. For any \(0<\varepsilon <1\) and \(s_0\ge 1\), there exist \(\ell =\ell (\Gamma )\in {\mathbb {N}}\) and \(\kappa =\kappa (\Gamma ,\varepsilon )>0\) satisfying: for every compact \(\Omega \subset G/\Gamma \) and \(\psi \in C_c^\infty (\Omega )\), there exists \(c = c(\Gamma ,{\text {supp}}\psi )\) such that for every \(x\in G/\Gamma \) that is \((\varepsilon ,s_0)\)-Diophantine, and for all \(r \gg _{\Gamma ,\Omega ,\varepsilon } s_0\),
where \(S_\ell (\psi )\) is the \(\ell \)-Sobolev norm.
3 Duality between \(G/\Gamma \) and \(U\backslash G\)
The goal of this section is to prove the following proposition, which shows that one can use equidistribution results of U orbits in \(G/\Gamma \) in order to study the distribution of the points in \(x\Gamma _T\) for \(x\in U\backslash G\). This will be used to prove Theorems 1.2 and 1.9 .
Recall that for \(x,y\in U\backslash G\), we defined
where \(E_{1,n+1}\) is the \((n+1)\times (n+1)\) matrix with one in the \((1,n+1)\)-entry and zeros everywhere else.
Recall the Iwasawa decomposition \(G={\text {SO}}(n,1)^\circ = U \times A \times K\). Define a continuous section by \(\Psi :U\backslash G\rightarrow AK\) by
where \(g=uak\) is the Iwasawa decomposition of g.
For \(\varphi \in C_c(U\backslash G)\), define
Proposition 3.1
Let \(\eta >0\), \(\Omega \subset U\backslash G\) be a compact set, \(\varphi \in C(\Omega )\), and \(\psi \in C(B_U(\eta ))\) be a non-negative function such that \(\int _U\psi =1\). Fix \(x \in U\backslash G.\) Define \(F\in C_c(G/\Gamma )\) by
Then, for some \(c=c(x,\Omega )>0\),
Observe that
is by definition the U-component of the Iwasawa decomposition of g. Similarly, for any \(g,h\in G\), the U-component of \(\Psi (Uh)g\) is given by
Hence, for any \(x\in U\backslash G\) and \(g\in G\), we can define \(c_x(g):=\mathbf{t }\in {\mathbb {R}}^{n-1}\), where \(\mathbf{t }\) is such that \(\Psi (x)g\Psi (xg)^{-1}=u_\mathbf{t }\). Then by the actions of U and A, we can see that this satisfies
and for any \(x\in U\backslash G\),
Observe that (20) implies that
Note that for \(g \in G\),
That is, \(u_{e}(g)\) is the U component of the Iwasawa decomposition of G, and \(\Psi (Ug)\) is the AK component.
For \(x \in U\backslash G\) and \(g \in G\), we will abuse notation and write
Lemma 3.2
For any compact \(\Omega \subset U\backslash G\) and \(x\in U\backslash G\) there exist \(c=c(\Omega ,x)>0\) such that for any \(xg \in \Omega \) and \(T>c\), we have
-
(1)
If \(\left\| g\right\| \le T\), then \(u_x(g)\in B_U\left( \frac{\sqrt{T}+c}{x\star xg}\right) \).
-
(2)
If \(\left\| g\right\| \ge T\), then \(u_x(g)\not \in B_U\left( \frac{\sqrt{T}-c}{x\star xg}\right) \).
Proof
We have \(g=\Psi (x)^{-1}u_x(g)\Psi (xg)\). For \(\mathbf{t }:=c_x(g)\) we get
Denote
Then, \(c_1\) and \(c_2\) are functions of x and \(\Omega \). By the triangle inequality,
In a similar way
We conclude that for any \(g\in \Omega \),
Assume \(\left\| g\right\| \ge T\ge c_1\). Then, by (22)
Using the quadratic formula, we may deduce that the right hand side of the above equation is equal to zero when
Since \((x\star xg)^2\) and \(\left\| t\right\| \) are non-negative, it follows that
Using the inequality \(\sqrt{a\pm b}\ge \sqrt{a}-\sqrt{b}\), we arrive at
A similar computation shows that \(\left\| g\right\| \le T\) implies
Letting c be the maximum of \(\frac{c_2+c_1x\star xg}{(x\star xg)^2}\) for \(g \in \Omega \) completes the proof. \(\square \)
Lemma 3.3
Let \(\varphi \in C_c(U\backslash G)\) and suppose that \(\psi \in C_c(U)\) satisfies
For \(g \in G\), define
Then for every \(g\in G\),
Proof
By the definition of \(\psi \),
Since \(\pi _U(u_{t}g)=\pi _U(g),\) we have
\(\square \)
We are now ready to prove Proposition 3.1.
Proof of Proposition 3.1
Without loss of generality, we may assume that \(\varphi \ge 0\). Define \(f:G\rightarrow {\mathbb {R}}\) by
By Lemma 3.3, for every \(g \in G\),
By Lemma 3.2, there exist \(c>0\) depending on \(\Omega \) and x such that for all \(T \ge c\), if \(\gamma \in \Gamma _T\) and \(x\gamma \in \Omega \), then
Observe also that since \({\text {supp}}(\psi )\subseteq B_U(\eta ),\) if \(u_\mathbf{t }\not \in u_x(\gamma )^{-1}B_U(\eta )\), then
Thus, using (21) and Lemma 3.3, for \(\gamma \in \Gamma _T\) with \(x\gamma \in \Omega \), we have that
Note that
Thus, from (23), for \(r=r_\varphi :=\min \limits _{y\in {\text {supp}}\varphi }(x\star y)\), we obtain
To obtain a lower bound, we must control the terms arising from \(\gamma \in \Gamma \setminus \Gamma _T\) in the definition of F. Note that by Lemma 3.2, if \(\gamma \in (\Gamma \setminus \Gamma _T)\) and \(x\gamma \in \Omega \), then we see that
Thus, by (23), we obtain
Now, similarly to the above, we conclude that
where \(R=R_\varphi :=\max \limits _{y \in {\text {supp}}\varphi }(x\star y),\) Then, the claim follows from the definition of F, (26). \(\square \)
Lemma 3.4
Let \(\varphi \in C_c(U\backslash G)\) and F be as defined in Proposition 3.1. Then,
Proof
By the definition of F and the assumption that \(\int _U\psi =1\), by the product structure of the BR measure in (14), we obtain
\(\square \)
For a set \(H \subseteq G\), let
where d is the Riemannian metric on G. That is, B(H, r) is the r-thickening of H with respect to d. For \(h\in G\), we denote \(B(\{h\},r)\) by B(h, r) (in this case we get the Riemannian ball around the point h).
For \(H \subseteq G\), denote by
the infimum over all \(r>0\) satisfying that for every \(h\in H,\)
is injective.
In the later sections, we will require a partition of \(\varphi \), say into \(\varphi _1,\ldots ,\varphi _k\) so that for each i, \(R_{\varphi _i}\) and \(r_{\varphi _i}\) are close.
Lemma 3.5
Fix \(x \in U\backslash G.\) For a compact set \(H \subseteq G,\) there exists \(0<\eta _0=\eta (H)<{\text {inj}}(H),\) \(\beta = \beta (H)>1\) so that for any \(0<\eta <\eta _0\) and \(\varphi \in C_c(U\backslash G)\) with \({\text {supp}}\varphi \subset \pi _U(B(h,\eta ))\) for some \(h \in H\), we have that
Proof
Since B(H, 1) is a compact set, by [4, Lemma 9.12], there exist constants \(0<\eta _0 = \eta (H)<{\text {inj}}(H)\), \(\beta =\beta (H)>1\), such that \(\eta _0<1\) and for all \(g,h \in B(H,1)\) with \(d(g,h)\le \eta _0\),
Therefore, for any \(h\in H\) and \(0<\eta <\eta _0\), we have
Note that for any \(g\in G\),
Thus, if \(\Vert g-h\Vert <\beta \eta \), then
and similarly
Thus, it follows from (17) that for
we have
Since r is bounded below by a constant depending on H, this implies that
\(\square \)
Corollary 3.6
Fix \(x \in U\backslash G\) and \(\varphi \in C_c(U\backslash G).\) Let \(\eta _0=\eta _0(\Psi ({\text {supp}}\varphi ))\) be as in Lemma 3.5. For any \(0<\eta <\eta _0,\) there exist some k and \(\varphi _1,\ldots ,\varphi _k \in C_c(U\backslash G)\) so that
Moreover, if \(\varphi \in C_c^\infty (U\backslash G)\), then we also have \(\varphi _i \in C_c^\infty (U\backslash G)\), and that for any \(\ell '>0\), there exists \(\ell > \ell '\) satisfying
Proof
For the first case (only assuming \(\varphi \in C_c(U\backslash G)\)), cover \(\Psi ({\text {supp}}\varphi )\) with balls of radius \(\eta ,\) and let \(\sigma _1,\ldots ,\sigma _k\) be a partition of unity subordinate to this cover. Defining
yields functions with the desired property, by Lemma 3.5.
Now, assume that \(\varphi \in C_c^\infty (U\backslash G)\), and let \(\ell >\ell '\) satisfy the conclusion of Lemma 2.12 for \(\ell '\). We must be more careful in order to control Sobolev norms. By Lemma 2.13, for \(0<\eta \le \eta _0\), there exist \(h_1,\ldots , h_k \in \Psi ({\text {supp}}\varphi )\) and \(\sigma _1,\ldots ,\sigma _k \in C_c^\infty (B(h_i,\eta ))\) with
and such that
Define
Then, by Lemma 3.5,
Since \(\Psi \) is smooth and \(\ell >\ell '\), by Lemmas 2.11 and 2.12,
From (31) and (32), we conclude that
\(\square \)
4 Proof of Theorem 1.2
In this section, we prove Theorem 1.2, which is restated below for convenience.
Theorem 4.1
Let \(\Gamma \) be geometrically finite. For any \(\varphi \in C_c(U\backslash G)\) and every \(x \in U\backslash G\) such that \(\Psi (x)^- \in \Lambda _r(\Gamma ),\)
We will need the following lemma. Theorem 1.2 will then follow by a partition of unity argument.
Lemma 4.2
Let \(\varphi \in C_c(U\backslash G)\) and let \(x\in U\backslash G\) be such that \(\Psi (x)^-\in \Lambda _r(\Gamma )\). Let \(R=R_\varphi \) and \(r=r_\varphi \) be as in (18). Let \(\eta >0\), and suppose that \(\frac{R}{r}<1+\eta \) and that \(B_U(\eta )\Psi ({\text {supp}}\varphi )\) injects into \(G/\Gamma \).
Then for any \(\varepsilon >0\), there exists \(T_1 = T_1(x,\eta ,\varphi )>0\) such that for all \(T \ge T_1\),
where \(\alpha =\alpha (\Gamma )\) is from Lemma 2.9, and \(c=c(x,{\text {supp}}\varphi )>0\) is as in Proposition 3.1.
Remark 4.3
Note that \(T_1\) depends on \(\eta \) through a non-canonical choice of bump function \(\psi \), as seen in the proof. When we apply this lemma to a partition of unity, the same \(\psi \) will be used for each part.
Proof
Let \(\psi \in C(B_U(\eta ))\) be a non-negative function such that \(\int _U \psi =1.\) Let F and \(c=c(x,{\text {supp}}\varphi )>0\) be as in the statement of Proposition 3.1 for this \(\psi \), and let \(\varepsilon >0\).
By Theorem 2.14, there exists \(T_1=T_1(x,\psi ,\varphi )\) such that for \(T \ge T_1\),
Let \(y \in {\text {supp}}(\varphi )\). By combining the above with Lemma 2.9 (using R, r, and \(\ell = x\star y\)), we see that there exist constants \(c_0=c_0(\Gamma ,x)\) and \(T_2 = T_2(\Gamma , x, {\text {supp}}\varphi )>0\) such that for \(T \ge T_2\),
By Lemma 3.4, \(m^{{\text {BR}}}(F)=\int _P \varphi (\pi _U(p))d\nu (p)\), and so by (36), for any \(y\in {\text {supp}}\varphi \), we obtain that
Since the above holds for any \(y \in {\text {supp}}\varphi ,\) by bounding
we obtain
\(\square \)
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2
By Corollary 3.6, there exists \(\eta _0=\eta _0(\Psi ({\text {supp}}\varphi ))>0\) so that for every \(0<\eta <\eta _0,\) there exists \(\left\{ \varphi _i\,:\,1\le i\le k\right\} \) that are a partition of \(\varphi \), i.e.,
so that all the \( \varphi _i\) are supported on a small neighborhood of \({\text {supp}}\varphi \), which we denote by B, and each \(\varphi _i\) satisfies the assumptions of Lemma 4.2.
For any \(1\le i\le k\) let,
as in (18).
Note that
satisfy \(R\ge R_i\ge r_i\ge r\) for any i.
Fix \(\varepsilon >0\). By Lemma 4.2, there exists \(T_1>0\) (depending on the \(\varphi _i\)’s, x, \(\eta \), and \(\varepsilon \)) such that for all \(T \ge T_1\) and for each i,
Summing over i, we obtain
Recall that
By Lemma 2.4, there exists \(\sigma =\sigma (\Gamma )>0\) so that for any \(y \in {\text {supp}}\varphi \),
Thus, from (37), we obtain
Since \(\eta \) and \(\varepsilon \) can be chosen arbitrarily small, the claim follows. \(\square \)
We will now deduce Corollary 1.3 using the shadow lemma, Proposition 2.1.
Proof of Corollary 1.3
Since \(\Psi (x)^- \in \Lambda _r(\Gamma )\), there exists \(r=r(x)\ge 0\) such that
Let \(w \in B_U(r)\Psi (x)\Gamma \cap {\text {supp}}m^{{\text {BMS}}}\subseteq G/\Gamma .\) Then for any \(T\ge 0\),
Thus, by Proposition 2.1, there exists \(\lambda =\lambda (\Gamma )>1\) such that for all \(T\ge 0\),
For every \(y \in {\text {supp}}\varphi ,\) we therefore have that for all \(T \ge 2r\),
By Theorem 1.2, there exists \(T_0=T_0(x,\varphi )\) such that for \(T \ge T_0\),
Then
so by (38), we obtain
The lower bound is very similar. \(\square \)
5 A small support “Ergodic Theorem”
In this section, we prove an ergodic-theorem type statement for functions with small support. This result will be used in the next chapter to prove Theorem 1.9.
Recall that for \(x\in U\backslash G\) and a compact set \(H\subset U\backslash G\), let
Theorem 5.1
Let \(\Gamma \) satisfy property A. There exists \(\ell =\ell (\Gamma ) \in {\mathbb {N}}\) so that for any \(0<\varepsilon <1\), there exists \(\kappa =\kappa (\Gamma ,\varepsilon )\) satisfying: for every \(x\in U\backslash G\) such that \(\Psi (x)\Gamma \) is \(\varepsilon \)-Diophantine and every compact \(\Omega \subset G\), there exists \(T_0=T_0(x,\Omega )\) so that for every \(T\ge T_0\), there exists \(\eta =\eta (T,\ell ,\kappa ,n,\Omega )>0\) such that if \(\varphi \in C^\infty _c(U\backslash G)\) with \(\Psi ({\text {supp}}\varphi )\subseteq \Omega \) and satisfies \({\mathcal {R}}({\text {supp}}\varphi ,x)-1<\eta \), then for every \(y\in {\text {supp}}\varphi \),
Proof
Fix \(x \in U\backslash G\) such that \(\Psi (x)\Gamma \) is \(\varepsilon \)-Diophantine. Let \(0<\eta _1=\eta _1(\Omega )<1\) be such that for all \(g \in \Omega \),
is injective, where \(B(g,\eta _1)=\{h \in G : \Vert g-h\Vert \le \eta _1\}\). Let \(0<\eta <\eta _1.\) Then if \(\Psi ({\text {supp}}\varphi )\subset \Omega \subset G\), we have that
injects into \(G/\Gamma \). Let \(R=R_\varphi \), \(r=r_\varphi \) as in (18). We are assuming that
We will find \(T_0=T_0(x,\Omega )\) as in the statement of the theorem, and choose \(\eta \) depending on \(T\ge T_0\) later.
According to Lemma 2.10(2), there exists \(\psi :U\rightarrow {\mathbb {R}}\) such that \({\text {supp}}\psi =B_U(\eta )\) and
We can now use Proposition 3.1 with the above \(\psi \) and \(\varphi \) to get an expression that we can estimate using the effective equidistribution theorem, Theorem 2.15.
Let F and \(c=c(\Omega ,x)\) be as in Proposition 3.1 for \(\psi ,\varphi \). There exists \(\ell ,\kappa ',c_2=c_2(\Gamma ,{\text {supp}}\psi ,x)\) as in the statement of Theorem 2.15 and \(T_1=T_1(x,\Omega )\ge c\) such that for all \(T \ge T_1\),
We now need to express \(m^{{\text {BR}}}(F)\) and \(S_\ell (F)\) in terms of \(\varphi \), and to compare the PS measures of the balls arising in (42) and (43).
Let \(y \in {\text {supp}}\varphi .\) Note that, by definition of r and R, \(r\le x\star y\le R\). Hence, we may use Lemma 2.9 to deduce that for \(r_-:=R\) and \(r_+:=r\), there exists \(T_2=T_2(x,\Omega )>0\) so that for all \(T\ge T_2\), we have that
According to Lemma 3.4, we have
Combining the above with (42) and (43) implies that, for some \(c_0=c_0(\Gamma ,x)\),
We are left to find \(S_\ell (F)\). Since \(B \mapsto B\Gamma \) is injective and f is supported on B (recall that f is defined as in Lemma 3.3), using Lemma 2.10(1), Lemma 2.12, and (41), we have
Finally, we need to put this all together. Combining (44) and (45), for any \(y\in {\text {supp}}\varphi \), we obtain that
Choose \(\rho \) sufficiently small so that
Let \(\eta =T^{-\rho }\), for \(T\ge T_0(x,\Omega ):=\max \{T_1,T_2\}.\) Let
Then we conclude that
\(\square \)
6 Proof of Theorem 1.9
In this section, we will use a partition of unity argument and the previous section to establish Theorem 1.9, which is restated below for convenience.
Theorem 6.1
Let \(\Gamma \) satisfy property A. There exists \(\ell =\ell (\Gamma )\in {\mathbb {N}}\) so that for any \(0<\varepsilon <1\), there exists \(\kappa =\kappa (\Gamma ,\varepsilon )\) satisfying: for every \(\varphi \in C^\infty _c(U\backslash G)\) and for every \(x\in U\backslash G\) such that \(\Psi (x)\Gamma \) is \(\varepsilon \)-Diophantine, and for all \(T\gg _{\Gamma ,{\text {supp}}{\varphi },x}1\),
Assume throughout this section that \(\Gamma \) satisfies property A. We begin by interpreting (46) in another form, as in the following lemma. This form will be easier to work with when using a partition of unity. Note that the main idea here is that for \(\varphi \) of small support and for any \(y\in {\text {supp}}\varphi \), \(x\star y\) is very close to both R and r.
For \(H \subseteq U\backslash G\) compact and \(x \in U\backslash G\), define
Lemma 6.2
There exists \(\ell =\ell (\Gamma )>0\) which satisfies the following. Let \(\Omega \subseteq G\) be a compact set, let \(x \in U \backslash G\) be such that \(\Psi (x)\Gamma \) is \(\varepsilon \)-Diophantine, let \(\varphi \in C_c^\infty (U\backslash G)\) with \(\Psi ({\text {supp}}\varphi )\subset \Omega \), and let \(\eta >0\) be smaller than the injectivity radius of \(\Omega \). Let \(R=R_{\pi _U(\Omega )}\) and \(r=r_{\pi _U(\Omega )}\) and assume they satisfy \(\frac{R}{r}-1<\eta .\) Then for \(T \gg _{\Gamma ,\Omega ,x} 1,\)
Proof
Following the arguments in the proof of Theorem 5.1 (more explicitly, the computations leading to (46)), one may deduce that there exists \(\ell =\ell (\Gamma )>0\) such that for any \(T \gg _{\Gamma ,\Omega ,x} 1,\)
Therefore, we may conclude
\(\square \)
Proof of Theorem 1.9
Step 1: Use an appropriate partition of \(\varphi \).
Let \(\ell '=\ell '(\Gamma )>0\) satisfy the conclusion of Lemma 6.2 and \(\ell >\ell '\) satisfy the conclusion of Corollary 3.6 for \(\ell '\).
By Corollary 3.6, there exists a partition \(\varphi _1,\ldots ,\varphi _k\) of \(\varphi \) satisfying Lemma 6.2 with \(\Omega = \Psi ({\text {supp}}\varphi )\) and
Thus, by Lemma 6.2, we have that for each \(\varphi _i\),
Let
Summing over i, using (47), and noting that \(\eta <1\) yields
Step 2: Putting it together.
Recall
Let
By Lemma 2.4, we have that there exists \(\sigma =\sigma (\Gamma )>0\) so that
where the last line follows because \((R/r)^\sigma \) is simply a constant depending on \({\text {supp}}\varphi ,\Gamma ,\) and x.
From (49) and (50), we obtain that
where (51) follows by choosing \(\eta =T^{-\rho }\), where \(\rho =1\) if \(2\ell -\frac{n^2+3n-2}{4}<0\), and
otherwise, and letting
\(\square \)
Remark 6.3
Note that the implied dependence on x is quite explicit. It arises from suppressing the factors \(R_\varphi , r_\varphi , \Vert \Psi (x)^{-1}\Vert ,\) and c throughout the argument. Specifically, c is suppressed in the use of Lemma 6.2, and \(r_{\varphi },R_\varphi \) are suppressed in (51). Note that these constants depend on x and \({\text {supp}}\varphi \) through the \(\star \) operation, as can be seen from the definitions and the proof of Lemma 3.2, and they can also be computed explicitly if desired. The factor of \(\Vert \Psi (x)^{-1}\Vert \) is suppressed in the construction of the partition in Corollary 3.6. The implied constant from Theorem 2.15 also depends on x through the explicit Diophantine behaviour of x, i.e. the \((\varepsilon ,s_0)\).
Remark 6.4
The suppressed constants \(R_\varphi ,r_\varphi , c,\) and \(\Vert \Psi (x)^{-1}\Vert \) mentioned in Remark 6.3 are continuous functions of x by definition of \(\star \). This will be used in the next section.
7 Applications
Let V be a manifold on which G acts smoothly and transitively from the right, so that V may be identified with \(H \backslash G\) for some closed subgroup H of G that is the stabilizer of a point \(v_0\in V\). Let \(\sigma : H \backslash G \rightarrow V\) be the identification
Note that \(\sigma \) is smooth because G acts smoothly.
Assume further that \(U \subseteq H \subseteq UM\). In particular, \(\pi _U(H)\) is compact in \(U\backslash G\) (recall from Sect. 2 that \(\pi _U : G \rightarrow U \backslash G\) is the quotient map). Define \(\theta : U \backslash G \rightarrow H \backslash G\) by
We will now show that \(\theta \) is smooth. Since U is closed, \(\pi _U : G \rightarrow U\backslash G\) is a smooth submersion. Thus, \(\theta \) is smooth if and only if \(\theta \circ \pi _U\) is smooth. Since \(\theta \circ \pi _U=\pi _H\), the quotient map from \(G \rightarrow H \backslash G\), it is smooth, which establishes the smoothness of \(\theta \).
For \(v, u \in V\), let \(x,y \in U\backslash G\) be such that \(u=\sigma (\theta (x)), v = \sigma (\theta (y))\). We may define
This is well-defined because UM stabilizes \(E_{1,n+1}\), and \(H \subseteq UM\) (see (1) for the definition of \(\star \) on \(U\backslash G\)).
Recall the definition of \(\Psi :U\backslash G \rightarrow G\) from Sect. 3:
where \(g=uak\) is the Iwasawa decomposition of g.
Definition 7.1
A vector \(v\in V\) is called \(\varepsilon \)-Diophantine if there exists \(x\in U\backslash G\) such that \(v=v_0 \cdot x\) and \(\Psi (x)\Gamma \) is \(\varepsilon \)-Diophantine. Such x is called an \(\varepsilon \)-Diophantine representative of v.
Remark 7.2
Note that for any \(g \in G\), \(g^-\in \Lambda _r(\Gamma )\) if and only if \((umg)^-\in \Lambda _r(\Gamma )\) for all \(um \in UM\), since UM does not change \(g^-\). Thus, for \(v \in V,\) we may define the notation
if for any representative \(\Psi (x),\) \(\Psi (x)^-\in \Lambda _r(\Gamma ).\) Note also that since \({\mathcal {C}}_0\) is M invariant and A commutes with M, the definition of v being \(\varepsilon \)-Diophantine is independent of the choice of a representative \(x\in U\backslash G\).
Observe that \(\nu \) uniquely defines a measure on \(U\backslash G\) by \(\nu (\varphi \circ \pi _{U})\) for any continuous function \(\varphi \) defined on \(U\backslash G\). One can use the push-forward of this measure to \(H\backslash G\) and the identification of V with \(H\backslash G\) to uniquely define a measure on V. Denote this measure by \({{\bar{\nu }}}\).
Corollary 7.3
For any \(0<\varepsilon <1\), there exist \(\ell =\ell (\Gamma )\in {\mathbb {N}}\) and \(\kappa =\kappa (\Gamma ,\varepsilon )\) satisfying: for every \(\overline{\varphi }\in C^\infty _c(V)\) and \(\varepsilon \)-Diophantine \(v\in V\) with Diophantine representative \(x\in U\backslash G\) (i.e., \(v_0x=v\)), and \(T\gg _{\Gamma ,{\text {supp}}{\overline{\varphi }},v}1\),
Proof
Let \(\ell '\) satisfy the conclusion of Theorem 1.9 and \(\ell \) satisfy the conclusion of Lemma 2.12 for \(\ell '\).
Recall the definitions of \(\sigma :H \backslash G \rightarrow V\) in (52) and \(\theta : U \backslash G \rightarrow H \backslash G\) in (53). Define \(\varphi \in C_c^\infty (U \backslash G)\) by
Let \(x \in U\backslash G\) be an \(\varepsilon \)-Diophantine representative of v. In particular, note that \(\sigma (\theta (x))=\sigma (H \Psi (x)) = v.\) Then, since
by Theorem 1.9, for \(T\gg _{\Gamma ,{\text {supp}}\overline{\varphi },\varepsilon ,x} 1\),
Note that the dependence of T on x is through \(\varepsilon ,s_0\) such that x is \((\varepsilon ,s_0)\)-Diophantine, and by Remark 7.2, this is in fact independent of the choice of Diophantine representative x of v. By Remark 6.4, the dependence on x in the implied constant in the above inequality can be made uniform over all representatives of v, as they vary by elements in M, a compact set. Thus, both dependencies on x can be replaced by dependence on v.
Observe that \(\varphi \) can be viewed as a function on \(U\backslash H \times H\backslash G\cong U\backslash G\) by
Therefore, Lemmas 2.10 and 2.12 imply
where the Sobolev norm of \({\text {id}}_{U\backslash H}\) is finite since we are assuming \(U\backslash H\) is compact. \(\square \)
In a similar way, one may deduce the following from Corollary 1.3 (see Remark 7.2 for the notation \(v^-\in \Lambda _r(\Gamma )\)):
Corollary 7.4
Assume that \(\Gamma \) is convex cocompact. For any \(\overline{\varphi }\in C_c(V)\) and every \(v\in V\) with \(v^-\in \Lambda _r(\Gamma )\), as \(T \rightarrow \infty \),
where the implied constant depends on v and \(\Gamma \).
7.1 Identification with null vectors
Let G act on \({\mathbb {R}}^{n+1}\) by right matrix multiplication, and let
To better understand the set V, note that the representation of \({\text {SO}}(n,1)\) we are using is
where
Let P be such that
Then VP is the upper half of the “light cone” in the standard representation of \({\text {SO}}(n,1)\). In particular, this consists of null vectors of
with \(x_{n+1}>0\). In our case, V consists of null vectors of
Proposition 7.5
Let \(\Gamma \) be convex cocompact. For any \(\overline{\varphi }\in C_c(V)\) and every \(v \in V\) with \(v^- \in \Lambda _r(\Gamma ),\) as \(T\rightarrow \infty \), we have that
where the implied constant depends on v and \(\Gamma \).
The measure \(\overline{\nu }\) is described more explicitly in (59), below.
Let
Then
and hence
via right matrix multiplication
We will now interpret Corollary 7.3 in this setting. We start by understanding the measure \(\overline{\nu }\).
We view V as \((M\backslash K)\times {\mathbb {R}}^+,\) via the “polar decomposition” of \(v \in V\),
where \({\mathbb {R}}^+ = \{r \in {\mathbb {R}}: r > 0\}\) and \(\Vert \cdot \Vert _2\) denotes the Euclidean norm on V. We may also identify \(M\backslash K\) with \(\partial ({\mathbb {H}}^{n})\) via
Thus, given \(v \in V\), (56) and (57) uniquely determine a pair \((a_{-\log \Vert v\Vert _2},Mk)\in A\times M\backslash K\), or equivalently, a pair \((a_{-\log \Vert v\Vert _2}, w_o^-k)\in A\times \partial ({\mathbb {H}}^n)\).
Viewing \(\partial ({\mathbb {H}}^n)\) as \(M\backslash K\) as in (57), we may in turn identify this with \({\mathbb {S}}^n \subseteq {\mathbb {R}}^{n+1}\) via
Thus, \(\nu _o\) uniquely determines a measure \(\overline{\nu }_o\) on \({\mathbb {S}}^n \cap V\) via
Then, since K stabilizes o and M stabilizes \(w_o\), \(\overline{\nu }\) can be described from (13): if \(s = \beta _{(a_{-\log \Vert v\Vert _2}k)^-}(o, a_{-\log \Vert v\Vert _2}k(o))=\log \left\| v\right\| _2\),
For \(v \in V\), define
where v corresponds to \((a_{-\log \Vert v\Vert _2},Mk) \in A \times M\backslash K.\) Then we have
As discussed in the previous section, \(v\star u\) may be computed by the formula in (1) for any choice of representatives of v and u in \(U\backslash G\). In particular, if
then
where \(k_{i,j}\) denotes the (i, j) entry of k. In particular
Putting this together with Corollary 7.4 yields the proposition.
7.2 Wedge products
The previous example can be generalized to \(\bigwedge ^j {\mathbb {R}}^{n+1}\) for any \(1\le j\le n\). Fix j, and let
with G acting on W by right multiplication. Then,
for some \(M_j\subseteq M\). Define
Fix a norm on V which is invariant under K such that \(\left\| v_0\right\| =1\).
Since any \(v\in V\) can be written as
where \(k\in M_j\backslash K\), in a similar way to the construction in the previous section, one can show that if \(a_{-\log \Vert v\Vert }k \in UP\) and can be written as \(uamv \in UAM{{\tilde{U}}}\), then
where \(v^-:=w_o^- k\), and dm is the push forward of the probability Haar measure on \(M_j\backslash M\). \(d\overline{\nu }(v)\) is zero if \(a_{-\log \Vert v\Vert }k\not \in UP\), because the original measure \(\nu \) is supported on P.
Moreover, by reasoning in the beginning of Sect. 7, \(v\star u\) is well defined and, as in the previous section, we have that
References
Aubin, T.: Nonlinear Analysis on Manifolds, vol. 252. Springer, New York (1982)
Bowditch, B.H.: Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113(2), 245–317 (1993)
Das, T., Fishman, L., Simmons, D., Urbański, M.: Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems. Ergod. Theory Dyn. Syst. (2021). https://doi.org/10.1017/etds.2020.46
Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259. Springer-Verlag London Ltd, London (2011)
Gorodnik, A.: Uniform distribution of orbits of lattices on spaces of frames. Duke Math. J. 122(3), 549–589 (2004)
Gorodnik, A., Nevo, A.: Ergodic theory and the duality principle on homogeneous spaces. Geom. Funct. Anal. 24, 159–244 (2014)
Gorodnik, A., Maucourant, F.: Proximality and equidistribution on the Furstenberg boundary. Geom. Dedicata 113, 197–213 (2005)
Gorodnik, A., Weiss, B.: Distribution of lattice orbits on homogeneous varieties. Geom. Func. An. 17, 58–115 (2007)
Kleinbock, D., Margulis, G.A.: Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinai’s Moscow Seminar on Dynamical Systems, American Mathematical Society Translations Series, 2, 171, American Mathematical Society, Providence, RI, pp. 141-172 (1996)
Ledrappier, F.: Distribution des orbites des réseaux sur le plan réel. C.R. Acad. Sci. Paris Sr. I Math. 329(1), 61–64 (1999)
Ledrappier, F., Pollicott, M.: Distribution results for lattices in \({{\rm SL}}_{2}({\mathbb{Q}}_p)\). Bull. Braz. Math. Soc. (N.S.) 36(2), 143–176 (2005)
Maucourant, F., Schapira, B.: Distribution of orbits in the plane of a finitely generated subgroup of \({{\rm SL}}(2,{\mathbb{R}})\). Am. J. Math. 136, 1497–1542 (2014)
Maucourant, F., Weiss, B.: Lattice actions on the plane revisited. Geometriae Dedicata 157, 1–21 (2012)
Mohammadi, A., Oh, H.: Classification of joinings for Kleinian groups. Duke Math. J. 165(11), 2155–2223 (2016)
Noguiera, A.: Orbit distribution on \({\mathbb{R}}^{2}\) under the natural action of \({\rm SL}(2,{\mathbb{Z}})\). Indag. Math. (N.S.) 13(1), 103–124 (2002)
Pollicott, M.: Rates of Convergence for Linear Actions of Cocompact Lattices on the Complex Plane, Integers, Volume 11B, Proceedings of the Leiden. Numeration Conference 2010 (2011)
Stratmann, B., Urbański, M.: Diophantine extremality of the Patterson measure. Math. Proc. Cambridge Philos. Soc. 140, 297–304 (2006)
Tamam, N., Warren, J. M.: Effective equidistribution of horospherical flows in infinite volume rank one homogeneous spaces, arXiv:2007.03135
Acknowledgements
We would like to thank Barak Weiss for bringing this problem to our attention, and Amir Mohammadi for many useful discussions. We are grateful to the anonymous referee for their insightful comments on an earlier version of this manuscript, which significantly improved the paper. The first author was partially supported by the Eric and Wendy Schmidt Fund for Strategic Innovation.
Open Access
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Tamam, N., Warren, J.M. Distribution of orbits of geometrically finite groups acting on null vectors. Geom Dedicata 216, 12 (2022). https://doi.org/10.1007/s10711-021-00669-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10711-021-00669-0