# Mapping Class Groups, Homology and Finite Covers of Surfaces

 Title: Mapping Class Groups, Homology and Finite Covers of Surfaces Author: Koberda, Thomas Citation: Koberda, Thomas. 2012. Mapping Class Groups, Homology and Finite Covers of Surfaces. Doctoral dissertation, Harvard University. Full Text & Related Files: Koberda_gsas.harvard_0084L_10228.pdf (556.8Kb; PDF) 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. Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10120819 Downloads of this work: