A note on the algebraic growth rate of Poincar\'e series for Kleinian groups

TL;DR

This paper uses infinite ergodic theory to estimate how quickly the Poincaré series of a Kleinian group grows at its critical exponent, providing insights into the group's geometric properties.

Contribution

It introduces a novel application of infinite ergodic theory to analyze the algebraic growth rate of Poincaré series for Kleinian groups at their critical exponent.

Findings

Derived estimates for the algebraic growth rate of Poincaré series.

Connected ergodic theory techniques with geometric group properties.

Provided bounds that improve understanding of Kleinian group dynamics.

Abstract

In this note we employ infinite ergodic theory to derive estimates for the algebraic growth rate of the Poincar\'e series for a Kleinian group at its critical exponent of convergence.