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 \begin{document}$ i^\text{th} $\end{document} and \begin{document}$ (i+1)^\text{th} $\end{document} Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index \begin{document}$ i $\end{document}. 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.