Aurel Page (University of Warwick):

Torsion homology of hyperbolic 3-manifolds in Jacquet-Langlands pairs and isospectral pairs

Abstract: I will first sketch an algorithm to compute fundamental domains, homology and Hecke operators for arithmetic Kleinian groups, including Bianchi groups. I will then describe ongoing work with Haluk Sengun where we numerically test and prove instances of the torsion Jacquet-Langlands conjecture, as formulated by Calegari and Venkatesh. I will explain the role played by "regulators" in the known results towards this conjecture, and present joint work with Alex Bartel on the behaviour of torsion homology and regulators in the less arithmetic but more tractable situation of Sunada's construction of isospectral manifolds.

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