On the growth spectrum of hyperbolic groups

Rémi Coulon; Michail Louvaris; Daniel T. Wise; Gal Yehuda

On the growth spectrum of hyperbolic groups

Group Theory 2026-07-02 v1

Authors: Rémi Coulon , Michail Louvaris , Daniel T. Wise , Gal Yehuda

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

Cite

@article{arxiv.2607.02147,
  title  = {On the growth spectrum of hyperbolic groups},
  author = {Rémi Coulon and Michail Louvaris and Daniel T. Wise and Gal Yehuda},
  journal= {arXiv preprint arXiv:2607.02147},
  year   = {2026}
}