Balanced ideals and domains of discontinuity of Anosov representations

Abstract

We consider Anosov subgroups of a semi-simple Lie group, a higher rank generalization of convex cocompact groups. Cocompact domains of discontinuity for these groups in flag manifolds were constructed systematically by Kapovich et al. (Geom Topol 22(1):157–234, 2018). For $\Delta$-Anosov groups, we show that every cocompact domain of discontinuity arises from this construction, up to a few exceptions in low rank. We use this to compute which flag manifolds admit such domains, and the number of different domains in some cases. We also find a new compactification for locally symmetric spaces arising from maximal representations into ${{\,\textrm{Sp}\,}}(4n+2,\mathbb {R})$.

A. Lists of balanced ideals

The description of cocompact domains using balanced ideals often reduces finding them to enumerating balanced ideals, which is a purely combinatorial problem. This means we can use a computer to do it. This section shows the resulting lists and numbers in some potentially interesting cases. The program used to compute them was written by the author together with David Dumas. It can be found online at https://florianstecker.net/balancedideals/.

1.1 A.1. Balanced ideals in $A_n$

Assume the Coxeter system $(W, \Delta )$ defined by the Weyl group and the restricted roots of G is of type $A_n$. For example, G could be the group ${{\,\textrm{SL}\,}}(n+1,\mathbb {R})$ or ${{\,\textrm{SL}\,}}(n+1,\mathbb {C})$. We write $\Delta = \{\alpha _1,\dots ,\alpha _n\}$ in such a way that $\alpha _i\alpha _{i+1} \in W$ is an element of order 3. The following tables show all balanced ideals in W. For every balanced ideal I, it shows the subset of $\Delta$ it is left- and right-invariant by. This means that for $\theta , \eta \subset \Delta$ the balanced ideals in $W_{\theta ,\eta }$ are precisely those in W whose left-invariance includes $\Delta \mathord \setminus \theta$ and right-invariance includes $\Delta \mathord \setminus \eta$. We also show the (real resp. complex) dimension of the set we have to take out of the ${\mathcal {F}}_\Delta$ for every limit point, and a minimal set of elements of W generating I as an ideal.

1.1.1 A.1.1. Balanced ideals in $A_1$

Left-invariance	Right-invariance	Dimension	Generators
$\varnothing$	$\varnothing$	0	1

1.1.2 A.1.2. Balanced ideals in $A_2$

Left-invariance	Right-invariance	Dimension	Generators
$\varnothing$	$\varnothing$	1	$\alpha _1, \alpha _2$

1.1.3 A.1.3. Balanced ideals in $A_3$

Left-invariance	Right-invariance	Dimension	Generators
$\{\alpha _1,\alpha _3\}$	$\{\alpha _1,\alpha _2\}$	4	$\alpha _3 \alpha _1 \alpha _2 \alpha _1$
$\{\alpha _1,\alpha _3\}$	$\{\alpha _2,\alpha _3\}$	4	$\alpha _1 \alpha _2 \alpha _3 \alpha _2$
$\{\alpha _2\}$	$\varnothing$	3	$\alpha _1\alpha _2\alpha _1,\; \alpha _2\alpha _1\alpha _3,\; \alpha _2\alpha _3\alpha _2$
$\varnothing$	$\{\alpha _1\}$	3	$\alpha _3\alpha _2\alpha _1,\; \alpha _1\alpha _2\alpha _1,\; \alpha _2\alpha _1\alpha _3$
$\varnothing$	$\{\alpha _2\}$	3	$\alpha _1\alpha _3\alpha _2,\; \alpha _1\alpha _2\alpha _1,\; \alpha _2\alpha _3\alpha _2$
$\varnothing$	$\{\alpha _3\}$	3	$\alpha _1\alpha _2\alpha _3,\; \alpha _2\alpha _1\alpha _3,\; \alpha _2\alpha _3\alpha _2$
$\varnothing$	$\varnothing$	3	$\alpha _3\alpha _2\alpha _1,\; \alpha _1\alpha _3\alpha _2,\; \alpha _1\alpha _2\alpha _3$
$\varnothing$	$\varnothing$	3	$\alpha _3\alpha _2\alpha _1,\; \alpha _1\alpha _2\alpha _3,\; \alpha _2\alpha _1\alpha _3$
$\varnothing$	$\varnothing$	3	$\alpha _3\alpha _2\alpha _1,\; \alpha _1\alpha _3\alpha _2,\; \alpha _1\alpha _2\alpha _1,\; \alpha _2\alpha _3$
$\varnothing$	$\varnothing$	3	$\alpha _1\alpha _3\alpha _2,\; \alpha _1\alpha _2\alpha _3,\; \alpha _2\alpha _3\alpha _2,\; \alpha _2\alpha _1$

1.1.4 A.1.4. The number of balanced ideals in $A_4$

There are 4608 balanced ideals in W, so we cannot list them all. Instead, the following table shows just how many balanced ideals exist in $W_{\theta ,\eta }$ for any choice of $\theta , \eta \subset \Delta$ with $\iota (\theta ) = \theta$. The rows correspond to different values of $\theta$ (for example 14 stands for $\theta = \{\alpha _1,\alpha _4\}$) while the columns correspond to $\eta$.

	1234	123	134	124	234	12	13	14	23	24	34	1	2	3	4
1234	4608	35	57	57	35	2	0	3	0	0	2	0	0	0	0
14	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
23	12	2	5	5	2	1	0	2	0	0	1	0	0	0	0

One feature stands out: There is only a single balanced ideal in $W_{\{\alpha _1,\alpha _4\},\Delta }$, and it has no right-invariances at all. In fact, we have the same situation generally in $W_{\{\alpha _1,\alpha _n\},\Delta }$ if W is of type $A_n$. For an $\{\alpha _1,\alpha _n\}$-Anosov representation into ${{\,\textrm{SL}\,}}(n+1,\mathbb {R})$ with limit map $\xi :\partial _\infty \Gamma \rightarrow {\mathcal {F}}_{1,n}$, this corresponds to the cocompact domain

$$\begin{aligned} \Omega = {\mathcal {F}}_\Delta \setminus \{F \in {\mathcal {F}}_\Delta \mid \exists x \in \partial _\infty \Gamma , i \le n :\xi ^{(1)}(x) \subset F^{(i)} \subset \xi ^{(n)}(x)\} \end{aligned}$$

which was also constructed in [16, 10.2.3] using the adjoint representation.

1.2 A.2. $\{\alpha _1,\dots ,\alpha _{p-1}\}$-Anosov representations into ${{\,\textrm{SO}\,}}_0(p,q)$

Guichard and Wienhard recently identified an interesting class of surface group representations they call $\Theta$-positive representations [17]. This includes a family of representations into ${{\,\textrm{SO}\,}}_0(p,q)$ with $p < q$ which they conjecture to be a union of connected components and to be $\theta$-Anosov with $\theta = \{\alpha _1,\dots ,\alpha _{p-1}\}$. Here we ordered the simple roots such that non-consecutive ones commute and $\alpha _{p-1}\alpha _p$ has order 4. If this conjecture is true, balanced ideals in $W_{\theta ,\eta }$ induce cocompact domains of discontinuity of these representations. Similarly to the table in Sect. 4.1, the following table shows the number of balanced ideals in $W_{\theta ,\eta }$ for $\eta = \{\alpha _k\}$, i.e. corresponding to domains in Grassmannians ${{\,\textrm{Is}\,}}_k(\mathbb {R}^{p,q})$ of isotropic k-subspaces.

	$k = 1$	$k = 2$	$k = 3$	$k = 4$	$k = 5$	$k = 6$	$k = 7$
$p = 2$	0	1
$p = 3$	0	1	1
$p = 4$	0	1	2	2
$p = 5$	0	1	7	14	3
$p = 6$	0	1	42	616	131	7
$p = 7$	0	1	429	303742	853168	8137	21

Notably, there is always a unique balanced ideal for $\eta = \{\alpha _2\}$. To give a more detailed characterization of this ideal, we describe elements $\xi \in {\mathcal {F}}_\theta$ as partial flags with subspaces $\xi ^i$ of dimensions i in the set $I {:}{=}\{1,2,\dots ,p-1\} \cup \{q+1,\dots ,p+q-1\}$, such that $\xi ^i$ is isotropic for $i < p$ and $\xi ^{p+q-i} = (\xi ^i)^\perp$. Here the “orthogonal complement” is taken with respect to the (p, q)-form. The relative positions of such a flag to a 2-dimensional isotropic plane $P \subset \mathbb {R}^{p+q}$ are fully described by the $2(p-1)$ numbers $d_i {:}{=}\dim (\xi ^i \cap P)$.

A non-decreasing tuple $(d_i)_{i \in I}$ of numbers $d_i \in \{0,1,2\}$ represents a valid relative position if and only if

$$\begin{aligned} d_1 \ne 2, \quad d_{p+q-1} \ne 0, \ \ \text {and} \ \ \textstyle \sum _{i\in I} d_i \ne 2(p-1). \end{aligned}$$

It is not hard to see that there are $2(p-1)^2$ valid tuples like this, and accordingly there are $2(p-1)^2$ relative positions. In this description, the unique balanced ideal consists of all relative positions described by tuples $(d_i)_{i\in I}$ with $\sum _{i\in I} d_i > 2(p-1)$.

Note that $\sum _{i \in I} d_i > 2(p-1)$ if and only if there is an index $i \le p-1$ with $d_i + d_{p+q-i} \ge 3$, equivalently $d_i \ne 0$ and $d_{p+q-i} = 2$. For the pair $(\xi ,P)$, this means that $\xi ^i \cap P \ne 0$ and $(\xi ^i)^\perp \cap P = P$ (or equivalently $P \perp \xi ^i$). Negating, we get that the relative position of $\xi$ and P is not in the balanced ideal if, for every index $i \le p-1$, either $\xi ^i \cap P = 0$ or $P \not \perp \xi ^i$. So if $\xi :\partial _\infty \Gamma \rightarrow {\mathcal {F}}_\theta$ is the boundary map of a $\theta$-Anosov representation (for example a $\Theta$-positive representation), then we can describe the unique cocompact domain of discontinuity in the space of isotropic planes as

$$\begin{aligned} \Omega = \{P \in {{\,\textrm{Is}\,}}_2(\mathbb {R}^{p,q}) \mid P \cap \xi ^{(i)}(x) = 0 \vee P \not \perp \xi ^{(i)}(x) \ \forall x \in \partial _\infty \Gamma \; \forall i \le p-1\}. \end{aligned}$$

1.3 A.3. $\{\alpha _1,\alpha _2\}$-Anosov representations into $F_4$

There is another exceptional family of $\Theta$-positive representations, which are conjectured to be $\{\alpha _1,\alpha _2\}$-Anosov in a group G with Weyl group of type $F_4$. This table shows the number of balanced ideals in $W_{\{\alpha _1,\alpha _2\},\eta }$ for different choices of $\eta$. Again, 134 is a shorthand for $\eta = \{\alpha _1,\alpha _3,\alpha _4\}$.

	1234	123	134	124	234	12	13	14	23	24	34	1	2	3	4
12	1270	182	140	66	44	16	18	5	14	6	4	1	2	2	0