Eigenvalue gaps for hyperbolic groups and semigroups

Abstract

<p style='text-indent:20px;'>Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the <inline-formula><tex-math id="M1">\begin{document}$ i^\text{th} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ (i+1)^\text{th} $\end{document}</tex-math></inline-formula> Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index <inline-formula><tex-math id="M3">\begin{document}$ i $\end{document}</tex-math></inline-formula>. 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.</p>