On Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$

TL;DR

This paper investigates the structure of Borel Anosov subgroups in SL(d,R), revealing they are virtually free or surface groups under certain conditions, and explores restrictions on hyperbolic space embeddings into symmetric spaces.

Contribution

It classifies Borel Anosov subgroups of SL(d,R) for specific dimensions, providing partial answers to Sambarino's question and analyzing hyperbolic space embeddings.

Findings

Borel Anosov subgroups are virtually free or surface groups for certain dimensions.

Restrictions are established on hyperbolic space embeddings into symmetric spaces.

The results depend on the dimension d and congruence conditions.

Abstract

We study the antipodal subsets of the full flag manifolds $\mathcal{F}(\mathbb{R}^d)$. As a consequence, for natural numbers $d \ge 2$ such that $d\ne 5$ and $d \not\equiv 0,\pm1 \mod 8$, we show that Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$ are virtually isomorphic to either a free group or the fundamental group of a closed hyperbolic surface. This gives a partial answer to a question asked by Andr\'es Sambarino. Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space $X_d$ of ${\rm SL}(d,\mathbb{R})$.