Group representations in the homology of 3-manifolds

Bartel, A. and Page, A. (2019) Group representations in the homology of 3-manifolds. Commentarii Mathematici Helvetici, 94(1), pp. 67-88. (doi:10.4171/CMH/455)

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Abstract

If M is a manifold with an action of a group G, then the homology group H1(M,Q) is naturally a Q[G]-module, where Q[G] denotes the rational group ring. We prove that for every finite group G, and for every Q[G]-module W, there exists a closed hyperbolic 3-manifold M with a free G-action such that the Q[G]-module H1(M,Q) is isomorphic to W. We give an application to spectral geometry: for every finite set P of prime numbers, there exist hyperbolic 3-manifolds N and N′ that are strongly isospectral such that for all p∈P, the p-power torsion subgroups of H1(N,Z) and of H1(N′,Z) have different orders. The main geometric techniques are Dehn surgery and, for the spectral application, the Cheeger–Müller formula, but we also make use of tools from different branches of algebra, most notably of regulator constants, a representation theoretic tool that was originally developed in the context of elliptic curves.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Bartel, Dr Alex
Authors: Bartel, A., and Page, A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Commentarii Mathematici Helvetici
Publisher:European Mathematical Society
ISSN:0010-2571
ISSN (Online):1420-8946
Published Online:05 March 2019
Copyright Holders:Copyright © 2018 Swiss Mathematical Society

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