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

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 $d\not\equiv 0,\pm 1\mathrm{mod}8$, we show that Borel Anosov subgroups of $\mathrm{SL}(d,\mathbb{R})$ 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 Andrés Sambarino. Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space $X_d$ of $\mathrm{SL}(d,\mathbb{R})$.

arXiv commentRevised after referee's comments. Accepted for publication in Geometry & Topology

References in corpus (6)

• Anosov representations: Domains of discontinuity and applications
• A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings
• Anosov representations and dominated splittings
• Topological restrictions on Anosov representations
• A combination theorem for Anosov subgroups
• Klein-Maskit combination theorem for Anosov subgroups: Free products

Cited by in corpus (2)

• Cubulated hyperbolic groups admit Anosov representations
• Topological and geometric restrictions on hyperconvex representations