Eigenvalue gaps for hyperbolic groups and semigroups

TL;DR

This paper investigates the relationship between Lyapunov exponent gaps and dominated splittings in hyperbolic groups and semigroups, extending concepts of Anosov representations to these algebraic structures.

Contribution

It establishes a connection between uniform Lyapunov exponent gaps and dominated splittings for hyperbolic groups and semigroups, introducing a new notion of Anosov representations in this context.

Findings

Uniform Lyapunov gap implies dominated splitting

Extension of Anosov representations to semigroups

Results for hyperbolic groups and subshifts

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-th and (i+1)-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.