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Convex cocompactness for Coxeter groups

Jeffrey Danciger, François Guéritaud, Fanny Kassel, Gye-Seon Lee, Ludovic Marquis

Abstract

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in arXiv:1704.08711.

Convex cocompactness for Coxeter groups

Abstract

We investigate representations of Coxeter groups into as geometric reflection groups which are convex cocompact in the projective space . We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in developed in arXiv:1704.08711.

Paper Structure

This paper contains 53 sections, 41 theorems, 55 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Convex cocompactness in projective space
  3. Vinberg's theory of reflection groups
  4. Coxeter groups admitting convex cocompact realizations
  5. A characterization of convex cocompactness
  6. The subspace of convex cocompact representations
  7. Anosov representations for Coxeter groups
  8. Organization of the paper
  9. Reminders: Actions on properly convex subsets of projective space
  10. Properly convex domains in projective space
  11. The proximal limit set and the full orbital limit set
  12. Maximal and minimal domains
  13. Convex cocompactness in projective space
  14. Reminders: Vinberg's theory for Coxeter groups
  15. Coxeter groups
  16. ...and 38 more sections

Key Result

Theorem 1.3

For an infinite Coxeter group $W_S$, the following are equivalent: In this case, $\rho(W_S)$ is actually convex cocompact in $\mathbb{P}(V)$ (not only naively convex cocompact), and we can take $V$ to be any vector space of dimension $\geqslant\# S$.

Figures (8)

  • Figure 1: Hilbert distance
  • Figure 2: Infinite Coxeter groups on two generators as in Example \ref{['ex:N=2']}, cases (i)--(ii)--(iii). The groups are shown acting on $\mathbb{R}^2$, $\mathbb{R}^2$ and $\mathbb{P}(\mathbb{R}^3)$ respectively. In the third panel, the points $[v_1]$, $[v_2]$ and $[u]$ are at infinity, and $\Omega_{\mathrm{TV}}$ is the full affine chart.
  • Figure 3: Illustration for the proof of Proposition \ref{['prop:maximal']}.\ref{['item:max-1']}
  • Figure 4: Illustration of Remark \ref{['rem:Vinberg-not-reduced']}.\ref{['item:inv-conv-not-in-OhmVin']} for $N=4$, where $V_v$ is a hyperplane in $V=\mathbb{R}^4$ and the affine reflections preserving the chart $\mathbb{P}(V)\smallsetminus \mathbb{P}(V_v) \simeq \mathbb{R}^3$ are chosen with linear parts in $\mathrm{O}(2,1)$. As above, $\Delta$ is the fundamental polytope for $\rho$, and $\Omega_{\mathrm{TV}}$ is the interior of $\bigcup_{\gamma\in W_S} \rho(\gamma)\cdot \Delta$. Here $\Omega_{\mathrm{TV}}$ intersects the chart in two connected components, each of which is a domain of dependence as in mes90 (see also dgk-proj-cc).
  • Figure 5: The sets $\Delta$, $\Sigma$, and $\Omega_{\mathrm{TV}}$ for $W_S:=\langle s_1, s_2, s_3 \,|\, s_i^2=1=(s_1 s_3)^2\rangle$ acting on $V=\mathbb{R}^3$ as a reflection group, preserving a copy of $\mathbb{H}^2$ in $\mathbb{P}(V)$. Here $(t_1,t_2,t_3)\mapsto t_1v_1+t_2v_2+t_3v_3$ gives coordinates on $V$. The set $\Delta$ (light gray) is a triangle bounded by the hyperplanes $\mathrm{Ker}(\alpha_i)$, and $\Sigma$ (dark gray) is the truncation of $\Delta$ by the hyperplanes $\{t_i=0\}$ (note that $\{t_2=0\}$ is at infinity). The set $\Omega_{\mathrm{TV}}$ is the interior of the $W_S$-orbit of $\Delta$ (here approximated by 8 iterates), and contains $\Sigma$.
  • ...and 3 more figures

Theorems & Definitions (113)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 103 more