Maximal symmetry groups of hyperbolic 3-manifolds

Every finite group acts as the full isometry group of some compact hyperbolic 3-manifold. In this paper we study those finite groups which act maximally, that is when the ratio |Isom(M)|/vol(M) is maximal among all such manifolds. In two dimensions maximal symmetry groups are called Hurwitz groups, and arise as quotients of the (2,3,7)–triangle group. Here we study quotients of the minimal co-volume lattice Γ of hyperbolic isometries in three dimensions, and its orientation-preserving subgroup Γ+, and we establish results analogous to those obtained for Hurwitz groups. In particular, we show that for every prime p there is some q = p such that either PSL(2, q) or PGL(2, q) is quotient of Γ+, and that for all but finitely many n, the alternating group An and the symmetric group Sn are quotients of Γ and also quotients of Γ +, by torsion-free normal subgroups. We also describe all torsion-free subgroups of index up to 120 in Γ+ (and index up to 240 in Γ), and explain how other infinite families of quotients of Γ and Γ+ can be constructed.