Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$\mathbb{H}^{p,q}$-convex cocompactness and higher higher Teichmüller spaces

Jonas Beyrer, Fanny Kassel

TL;DR

The paper develops a higher-dimensional, pseudo-Riemannian generalization of convex cocompact and Teichmüller-type phenomena by introducing $ H^{p,q}$-convex cocompact representations into ${ m PO}(p,q+1)$. It relates these representations to Anosov dynamics, weakly spacelike $p$-graphs, and non-positive boundary spheres, establishing that for groups with maximal virtual cohomological dimension $p$ the Hpq-cc representations form unions of connected components in ${ m Hom}( Gamma,{ m PO}(p,q+1))$, thereby producing higher-dimensional higher Teichmüller spaces. The work also proves non-degeneracy stability under reductive limits, existence of invariant weakly spacelike graphs, and crowns-geometry-based obstructions to hyperbolicity, culminating in closedness results and new examples including non-hyperbolic groups and Zariski-dense deformations. It significantly broadens the class of groups and ambient geometries supporting higher-rank Teichmüller-type phenomena, with potential implications for geometric group theory and higher-rank discrete subgroups. The methods intertwine convex projective geometry, Anosov theory, and Lorentzian/AdS-type constructions to produce a robust framework for higher-dimensional Teichmüller spaces in indefinite orthogonal groups.

Abstract

For any integers $p\geq 2$ and $q\geq 1$, let $\mathbb{H}^{p,q}$ be the pseudo-Riemannian hyperbolic space of signature $(p,q)$. We prove that if $Γ$ is the fundamental group of a closed aspherical $p$-manifold, then the set of representations of $Γ$ to $\mathrm{PO}(p,q+1)$ which are convex cocompact in $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(Γ,\mathrm{PO}(p,q+1))$. More generally, we show that if $Γ$ is any finitely generated group with no infinite nilpotent normal subgroups and with virtual cohomological dimension $p$, then the set of injective and discrete representations of $Γ$ to $\mathrm{PO}(p,q+1)$ preserving a non-degenerate non-positive $(p-1)$-sphere in the boundary of $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(Γ,\mathrm{PO}(p,q+1))$. This gives new examples of higher-dimensional higher-rank Teichmüller spaces.

$\mathbb{H}^{p,q}$-convex cocompactness and higher higher Teichmüller spaces

TL;DR

The paper develops a higher-dimensional, pseudo-Riemannian generalization of convex cocompact and Teichmüller-type phenomena by introducing -convex cocompact representations into . It relates these representations to Anosov dynamics, weakly spacelike -graphs, and non-positive boundary spheres, establishing that for groups with maximal virtual cohomological dimension the Hpq-cc representations form unions of connected components in , thereby producing higher-dimensional higher Teichmüller spaces. The work also proves non-degeneracy stability under reductive limits, existence of invariant weakly spacelike graphs, and crowns-geometry-based obstructions to hyperbolicity, culminating in closedness results and new examples including non-hyperbolic groups and Zariski-dense deformations. It significantly broadens the class of groups and ambient geometries supporting higher-rank Teichmüller-type phenomena, with potential implications for geometric group theory and higher-rank discrete subgroups. The methods intertwine convex projective geometry, Anosov theory, and Lorentzian/AdS-type constructions to produce a robust framework for higher-dimensional Teichmüller spaces in indefinite orthogonal groups.

Abstract

For any integers and , let be the pseudo-Riemannian hyperbolic space of signature . We prove that if is the fundamental group of a closed aspherical -manifold, then the set of representations of to which are convex cocompact in is a union of connected components of . More generally, we show that if is any finitely generated group with no infinite nilpotent normal subgroups and with virtual cohomological dimension , then the set of injective and discrete representations of to preserving a non-degenerate non-positive -sphere in the boundary of is a union of connected components of . This gives new examples of higher-dimensional higher-rank Teichmüller spaces.
Paper Structure (56 sections, 63 theorems, 61 equations, 1 figure, 1 table)

This paper contains 56 sections, 63 theorems, 61 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Previously known higher higher Teichmüller spaces and new examples
  3. $\mathbb H^{p,q}$-convex cocompact representations
  4. Link with Anosov representations
  5. Weakly spacelike $p$-graphs with proper cocompact group actions
  6. A non-degeneracy result
  7. Existence of invariant weakly spacelike $p$-graphs
  8. Higher higher Teichmüller spaces for non-hyperbolic groups
  9. Outline of the proofs of Theorem \ref{['thm:geom-action-weakly-sp-gr-basic']} and Proposition \ref{['prop:non-deg-limit']}
  10. Proof of Theorem \ref{['thm:geom-action-weakly-sp-gr-basic']}
  11. Proof of Proposition \ref{['prop:non-deg-limit']}
  12. Organization of the paper
  13. Reminders
  14. Properly convex domains in projective space
  15. The pseudo-Riemannian hyperbolic space $\mathbb H^{p,q}$ and its boundary at infinity
  16. ...and 41 more sections

Key Result

Theorem 1.1

Let $p\geq 2$ and $q\geq 1$ be integers. Let $N$ be a closed real hyperbolic $p$-manifold with holonomy representation $\sigma_0 : \pi_1(N)\to{\rm PO}(p,1)$, and let $\rho_0 : \pi_1(N)\to{\rm PO}(p,q+1)$ be the composition of a lift of $\sigma_0$ to ${\rm O}(p,1)$ with the natural inclusion ${\rm O}

Figures (1)

  • Figure 1: The anti-de Sitter space $\mathrm{AdS}^3 = \mathbb H^{2,1}$, intersected with an affine chart of $\mathbb P(\mathbb R^4)$, is the interior of the quadric of equation $v_1^2 + v_2^2 - v_3^2 = 0$. Here we have represented, in such an affine chart, the image $M$ in $\mathbb H^{2,1}$ of a spacelike $2$-graph as in Example \ref{['ex:spacelike-graph']}.(ii). Its ideal boundary is the union of four segments of $\partial_{\infty}\mathbb H^{2,1}$, joining points $x_1^{\pm}$ and $x_2^{\pm}$.

Theorems & Definitions (149)

  • Theorem 1.1
  • Definition 1.2: dgk18
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 139 more