Group representations in the homology of 3-manifolds

Abstract

If $M$ is a manifold with an action of a group $G$, then the homology group $H_1(M,\mathbb{Q})$ is naturally a $\mathbb{Q}[G]$-module, where $\mathbb{Q}[G]$ denotes the rational group ring. We prove that for every finite group $G$, and for every $\mathbb{Q}[G]$-module $W$, there exists a closed hyperbolic 3-manifold $M$ with a free $G$-action such that the $\mathbb{Q}[G]$-module $H_1(M,\mathbb{Q})$ is isomorphic to $W$. We give an application to spectral geometry: for every finite set $\mathcal{P}$ of prime numbers, there exist hyperbolic 3-manifolds $N$ and $N^{\prime }$ that are strongly isospectral such that for all $p\in \mathcal{P}$, the $p$-power torsion subgroups of $H_1(N,\mathbb{Z})$ and of $H_1(N^{\prime },\mathbb{Z})$ 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.