Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity

TL;DR

This paper studies the Hausdorff dimension of limit sets of irreducible projective Anosov subgroups, revealing conditions under which the dimension is full, partial, or computable via self-affine properties.

Contribution

It establishes new links between the Hausdorff dimension of limit sets and the algebraic and geometric properties of Anosov subgroups, including self-affine complexity and regular distortion.

Findings

Full Hausdorff dimension implies $d=2$ and cocompact lattice.

Limit sets of surface groups in $d=3$ are not of dimension 1 unless Hitchin.

Partial quasi-self-similarity allows explicit dimension calculation.

Abstract

Let $\Gamma\subset \mathsf{PGL}(d,\mathbb{R})$ be an irreducible projective Anosov subgroup and let $\Lambda^1(\Gamma)$ be its projective limit set. Viewing $\Lambda^1(\Gamma)$ as an analogue of a self-affine set, we investigate the Hausdorff dimension of $\Lambda^1(\Gamma)$ under specific assumptions regarding its affine complexity: 1. If $\Lambda^1(\Gamma)$ is of full Hausdorff dimension, then $d= 2$ and $\Gamma$ is a cocompact lattice. 2. If $d = 3$ and $\Gamma$ is the image of a closed surface group under an irreducible Anosov representation, then $\Lambda^1(\Gamma)$ never has Hausdorff dimension $1$ unless the representation is Hitchin. 3. If the limit set $\Lambda^1(\Gamma)$ exhibits a partial quasi-self-similarity (in the sense of Falconer~\cite{falconerselfsimilar1}) -- which can be implied by the ``regular distortion property'' of $\Gamma$ -- then the Hausdorff dimension of $\Lambda^1(\Gamma)$ equals the critical exponent of the first simple root. An application of this result is the computation of the Hausdorff dimension of the limit set for arbitrary $\Theta$-positive representations of convex cocompact Fuchsian groups.