Mapping Class Groups, Homology and Finite Covers of Surfaces

Abstract

Let S be an orientable surface of genus g with n punctures, such that \(\chi(S) = 2 − 2g − n < 0\). Let \(\psi \epsilon Mod(S)\) denote an element in its mapping class group. In this thesis, we study the action of \(\psi\) on \(H_1(\tilde{S}, \mathbb{C})\), where \(\tilde{S}\)varies over the finite covers of S to which \(\psi\) lifts. We first show that if \(\psi\)is a nontrivial mapping class then there exists a finite cover \(\tilde{S}\) such that each lift of \(\psi\) to \(\tilde{S}\) acts nontrivially on \(H_1(\tilde{S}, \mathbb{C})\). We then show that the combination of the lifted actions of \(\psi\) and the Galois groups of the covers on \(H_1(\tilde{S}, \mathbb{C})\) can be used to determine the Nielsen–Thurston class of \(\psi\). We then turn to the following conjecture: that for each pseudo-Anosov mapping class \(\psi\), there exists a lift \(\tilde{\psi}\) to a finite cover whose action on \(H_1(\tilde{S}, \mathbb{C})\) has spectral radius strictly greater than one. We show that the conjecture holds if and only if the mapping torus \(T_{\psi}\) has exponential growth of torsion homology with respect to a particular collection of finite covers. We use growth of torsion homology to characterize the mapping classes for which the conjecture holds. Then, we show that if the conjecture fails then \(T_{\psi}\) is large.