Volume entropy for minimal presentations of surface groups in all ranks

Abstract

Agraïments: This work has been carried out thanks to the support of the ARCHIMEDE Labex (ANR-11-LABX- 0033).

We study the volume entropy of a class of presentations (including the classical ones) for all surface groups, called minimal geomètric presentations. We rediscover a formula first obtained by Cannon and Wagreich [6] with the computation in a non published manuscrit by Cannon [5]. 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 [3] and extended to all geometric presentations in [15]. 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 Gn of rank n with a classical presentation Pn the volume entropy is log(λn), where λn is the unique real root larger than one of the polynomial x n − 2(n − 1) nX−1 j=1 x j + 1.