Eigenvalue gaps for hyperbolic groups and semigroups

Abstract

Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the $ i^\text{th} $ and $ (i+1)^\text{th} $ Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index $ i $. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.

Topics & Concepts