The H\"older exponent of Anosov limit maps

TL;DR

This paper computes the Hölder exponent of Anosov limit maps for hyperbolic groups, linking it to eigenvalue moduli and translation lengths, and explores cases where the exponent is attained or not.

Contribution

It provides explicit formulas for the Hölder exponent of Anosov limit maps in terms of eigenvalues and translation lengths, including new examples and cases where the exponent is not attained.

Findings

Explicit formula for Hölder exponent in terms of eigenvalues and translation lengths.

Conditions under which the limit map attains its Hölder exponent.

Examples of non semisimple representations where the exponent is not attained.

Abstract

Let $\Gamma$ be a non-elementary word hyperbolic group and $d_{a}, a>1,$ a visual metric on its Gromov boundary $\partial_{\infty}\Gamma$. For an $1$-Anosov representation $\rho:\Gamma \rightarrow \mathsf{GL}_{d}(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the H\"older exponent of the Anosov limit map $\xi_{\rho}^1:(\partial_{\infty}\Gamma, d_{a})\rightarrow (\mathbb{P}(\mathbb{K}^d),d_{\mathbb{P}})$ of $\rho$ in terms of the moduli of eigenvalues of elements in $\rho(\Gamma)$ and the stable translation length on $\Gamma$. If $\rho$ is either irreducible or $\xi_{\rho}^1(\partial_{\infty}\Gamma)$ spans $\mathbb{K}^d$ and $\rho$ is $\{1,2\}$-Anosov, then $\xi_{\rho}^1$ attains its H\"older exponent. We also provide an analogous calculation for the exponent of the inverse limit map of $(1,1,2)$-hyperconvex representations. Finally, we exhibit examples of non semisimple $1$-Anosov representations of surface groups in $\mathsf{SL}_4(\mathbb{R})$ whose Anosov limit map in $\mathbb{P}(\mathbb{R}^4)$ does not attain its H\"older exponent.