On the growth spectrum of hyperbolic groups

Abstract

We study the growth spectrum of groups acting on hyperbolic spaces, i.e.\ the set of exponential growth rates achieved by subgroups. For a finitely generated free group or a surface group acting convex-cocompactly on a proper geodesic hyperbolic metric space, we prove that the growth spectrum is the full interval $[0,\ddot{I}{‰}_G]$. For any hyperbolic group, we prove that the growth spectrum contains a large interval $[0,\ddot{I}{‰}_{\mathcal{F}}]$ where $\ddot{I}{‰}_{\mathcal{F}}\ge \ddot{I}{‰}_G/2$, with strict inequality when the action is divergent. In the case of the Cayley graph of a free group, we also present an approach via the non-backtracking matrix of the configuration model, connecting the density of growth rates to a spectral concentration result for random graphs.