Volume entropy for surface groups via Bowen-Series like maps

TL;DR

This paper introduces a Bowen-Series like map for geometric presentations of surface groups, linking volume entropy to topological entropy, and identifies the minimal volume entropy among such presentations.

Contribution

It defines a new circle map for surface groups and establishes a direct relation between volume and topological entropy, also finding the minimal volume entropy.

Findings

Volume entropy equals topological entropy of the constructed map

A method to compute volume entropy via circle maps

Identification of minimal volume entropy among geometric presentations

Abstract

We define a Bowen-Series like map for every geometric presentation of a co-compact surface group and we prove that the volume entropy of the presentation is the topological entropy of this particular (circle) map. Finally we find the minimal volume entropy among geometric presentations.