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 $[0, ω_G]$. For any hyperbolic group, we prove that the growth spectrum contains a large interval $[0, ω_{\mathcal{F}}]$ where $ω_{\mathcal{F}} \geq ω_G / 2$, 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.