Continuity of critical exponent of quasiconvex-cocompact groups under Gromov-Hausdorff convergence

TL;DR

This paper proves that the critical exponent of certain hyperbolic groups remains continuous under Gromov-Hausdorff convergence, ensuring stability of this geometric invariant in a broad class of hyperbolic spaces.

Contribution

It establishes the continuity of the critical exponent for quasiconvex-cocompact groups under equivariant Gromov-Hausdorff convergence, extending understanding of geometric invariants in hyperbolic geometry.

Findings

Critical exponent is continuous under Gromov-Hausdorff convergence.

Continuity holds for groups with uniformly bounded codiameter.

Results apply to non-elementary, torsion-free hyperbolic groups.

Abstract

We show continuity under equivariant Gromov-Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces.