Purely exponential growth of cusp-uniform actions

TL;DR

This paper characterizes when a group acting on a hyperbolic space exhibits purely exponential orbit growth, linking it to a specific condition and extending known results to geometrically finite manifolds.

Contribution

It provides a new characterization of purely exponential growth for cusp-uniform actions using Patterson-Sullivan measures and a variant of the shadow lemma, extending prior theorems.

Findings

Pure exponential growth is equivalent to Dal'bo-Otal-Peigné condition.

Finiteness of Bowen-Margulis-Sullivan measures for certain manifolds.

Extension of Roblin's theorem to broader settings.

Abstract

Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal'bo-Otal-Peign\'e. For geometrically finite Cartan-Hadamard manifolds with pinched negative curvature this condition ensures the finiteness of Bowen-Margulis-Sullivan measures. In this case, our result recovers a theorem of Roblin (in a weaker form). Our main tool is the Patterson-Sullivan measures on the Gromov boundary of $X$, and a variant of the Sullivan shadow lemma called partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the condition of Dal'bo-Otal-Peign\'e. These results are further used in the paper \cite{YANG7}.