Anosov subgroups: Dynamical and geometric characterizations

TL;DR

This paper explores Anosov subgroups in higher rank Lie groups, providing new characterizations that generalize convex cocompactness from rank one, simplifying definitions, and relaxing conditions for broader applicability.

Contribution

It introduces several new equivalent characterizations of Anosov subgroups, simplifying the original definition and extending the concept with relaxed conditions.

Findings

New characterizations capture 'rank one behavior' in higher rank groups.

Simplified the original definition by avoiding geodesic flow.

Extended the Anosov condition with non-uniform unbounded expansion.

Abstract

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture "rank one behavior" of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.