Volume entropy for minimal presentations of surface groups in all ranks

TL;DR

This paper derives an explicit polynomial formula for the volume entropy of minimal geometric presentations of surface groups across all ranks, using a dynamical systems approach inspired by Bowen and Series.

Contribution

It introduces a new dynamical systems method to compute volume entropy for all surface group presentations, revealing a polynomial dependence on the group rank.

Findings

Explicit polynomial formula for volume entropy in terms of group rank

Volume entropy equals the logarithm of the polynomial's largest root

Method applies to all minimal geometric presentations of surface groups

Abstract

We study the volume entropy of a class of presentations (including the classical ones) for all surface groups, called \emph{minimal geometric presentations}. We rediscover a formula first obtained by Cannon and Wagreich with the computation in a non published manuscript by Cannon. The result is surprising: an explicit polynomial of degree $n$, the rank of the group, encodes the volume entropy of all classical presentations of surface groups. The approach we use is completely different. It is based on a dynamical system construction following an idea due to Bowen and Series and extended to all geometric presentations in. The result is an explicit formula for the volume entropy of minimal presentations for all surface groups, showing a polynomial dependence in the rank $n>2$. We prove that for a surface group $G_n$ of rank $n$ with a classical presentation $P_n$ the volume entropy is $\log(\lambda_n)$, where $\lambda_n$ is the unique real root larger than one of the polynomial \[ x^{n} - 2(n - 1) \sum_{j=1}^{n-1} x^{j} + 1. \]