Volume entropy for surface groups via Bowen-series-like maps (Q2879006)

This interesting paper studies the volume entropy \(h(G,X,R)\) for a finitely presented group \(G\) with a finite presentation \(\langle X,R \rangle\) (\(X\) denotes the set of generators of \(G\) and \(R\) the relations), giving rise to a metric space \((G, d_{X})\) with the word metric \(d_{X}\). This entropy is defined by NEWLINE\[NEWLINE h(G,X,R):= \lim_{n \to \infty} \frac{1}{n} \ln |S(n)|_{X},NEWLINE\]NEWLINE where \( |S(n)|_{X}\) denotes the cardinality of the sphere of radius \(n\) in \((G, d_{X})\).NEWLINENEWLINEThe main results of the paper are stated in the following two theorems. The proofs of these theorems are totally geometric and constructive.NEWLINENEWLINETheorem 1. Let \(G\) be a co-compact hyperbolic surface group with a geometric presentation \(\langle X,R \rangle\). Then there exists a Markov map \(\phi : \partial G = {\mathbb S}^{1} \to \partial G\) which is orbit equivalent to the group action. For this particular map we have that \(h(G,X,R)\) is equal to the topological entropy of the map \(\phi\).NEWLINENEWLINETheorem 2. The minimal volume entropy of a hyperbolic surface group, among all geometric presentations, is realised by the presentations with the minimal number of generators. In particular, this minimum volume entropy is explicitly computable.