Arithmetic Kleinian groups are arithmetic lattices in PSL 2 (C). We present an algorithm that, given such a group Γ, returns a fundamental domain and a finite presentation for Γ with a computable isomorphism. 2

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Abstract. Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M1 and M2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. Here, we investigate the relationship between their integral homology. The Cheeger-Müller Theorem implies that a certain product of orders of torsion homology and of regulators for M1 agrees with that for M2. We exhibit a connection between the torsion in the integral homology of M1 and M2 on the one hand, and the G-module structure of integral homology of the covering manifold on the other, by interpreting the quotients Reg i (M1)/ Reg i (M2) representation theoretically. Further, we prove that the p ∞ -torsion in the homology of M1 is isomorphic to that of M2 for all primes p #G. For p ≤ 71, we give examples of pairs of strongly isospectral hyperbolic 3-manifolds for which the p-torsion homology differs, and we conjecture such examples to exist for all primes p.

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.