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Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps

Konstantinos Tsouvalas

TL;DR

This paper develops new characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups by relating boundary limit maps, the Cartan projection, and the uniform gap summation property. It shows that Anosov representations can be detected from the existence of limit maps with Cartan-type control, and that weak uniform eigenvalue gaps together with additional dynamics yield Anosovness, with consequences for strongly convex cocompact subgroups in $\mathsf{PGL}_d(\mathbb{R})$. It also connects these representations to geometric actions on properly convex domains and the Hilbert metric, providing a geometric/dynamical criterion for strong convex cocompactness. The results unify several prior characterizations (GGKW, KLP, BPS, DGK0) and provide new tools for constructing and recognizing Anosov and convex cocompact groups, including in Floyd-boundary settings.

Abstract

We provide characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups in terms of the existence of equivariant limit maps on the Gromov boundary, the Cartan property and the uniform gap summation property introduced by Guichard-Guéritaud-Kassel-Wienhard. We also study representations of finitely generated groups satisfying weak uniform gaps in eigenvalues and establish conditions to be Anosov. As an application, we also obtain a characterization of strongly convex cocompact subgroups of the projective linear group $\mathsf{PGL}_d(\mathbb{R})$.

Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps

TL;DR

This paper develops new characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups by relating boundary limit maps, the Cartan projection, and the uniform gap summation property. It shows that Anosov representations can be detected from the existence of limit maps with Cartan-type control, and that weak uniform eigenvalue gaps together with additional dynamics yield Anosovness, with consequences for strongly convex cocompact subgroups in . It also connects these representations to geometric actions on properly convex domains and the Hilbert metric, providing a geometric/dynamical criterion for strong convex cocompactness. The results unify several prior characterizations (GGKW, KLP, BPS, DGK0) and provide new tools for constructing and recognizing Anosov and convex cocompact groups, including in Floyd-boundary settings.

Abstract

We provide characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups in terms of the existence of equivariant limit maps on the Gromov boundary, the Cartan property and the uniform gap summation property introduced by Guichard-Guéritaud-Kassel-Wienhard. We also study representations of finitely generated groups satisfying weak uniform gaps in eigenvalues and establish conditions to be Anosov. As an application, we also obtain a characterization of strongly convex cocompact subgroups of the projective linear group .

Paper Structure

This paper contains 22 sections, 28 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Characterizations in terms of limit maps and the Cartan property
  3. Weak uniform gaps in eigenvalues and strong property $\textup{(U)}$.
  4. Characterizations of strongly convex cocompact groups
  5. Gromov product.
  6. Background
  7. Lie theory
  8. Gromov hyperbolic spaces.
  9. The Floyd boundary.
  10. Flow spaces for hyperbolic groups.
  11. Anosov representations
  12. Semisimple representations.
  13. Convex cocompact groups
  14. The contraction property
  15. The Cartan property and the uniform gap summation property
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a word hyperbolic group, $G$ a real semisimple Lie group, $\theta \subset \Delta$ a subset of simple restricted roots of $G$ and $\rho:\Gamma \rightarrow G$ a representation. Then $\rho$ is $P_{\theta}$-Anosov if and only if the following conditions are simultaneously satisfied:

Theorems & Definitions (63)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 53 more