Group representations in the homology of 3-manifolds

  • Alex Bartel

    University of Glasgow, UK
  • Aurel Page

    Université de Bordeaux, Talence, France
Group representations in the homology of 3-manifolds cover
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If MM is a manifold with an action of a group GG, then the homology group H1(M,Q)H_1(M,\mathbb Q) is naturally a Q[G]\mathbb Q[G]-module, where Q[G]\mathbb Q[G] denotes the rational group ring. We prove that for every finite group GG, and for every Q[G]\mathbb Q[G]-module WW, there exists a closed hyperbolic 3-manifold MM with a free GG-action such that the Q[G]\mathbb Q[G]-module H1(M,Q)H_1(M,\mathbb Q) is isomorphic to WW. We give an application to spectral geometry: for every finite set P\mathcal P of prime numbers, there exist hyperbolic 3-manifolds NN and NN' that are strongly isospectral such that for all pPp \in \mathcal P, the pp-power torsion subgroups of H1(N,Z)H_1(N,\mathbb Z) and of H1(N,Z)H_1(N',\mathbb 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.

Cite this article

Alex Bartel, Aurel Page, Group representations in the homology of 3-manifolds. Comment. Math. Helv. 94 (2019), no. 1, pp. 67–88

DOI 10.4171/CMH/455