Mapping Class Groups, Homology and Finite Covers of Surfaces

DSpace/Manakin Repository

Mapping Class Groups, Homology and Finite Covers of Surfaces

Citable link to this page

 

 
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:
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:

Show full Dublin Core record

This item appears in the following Collection(s)

 
 

Search DASH


Advanced Search
 
 

Submitters