Abstract
A finite group is called a Hurwitz group if it is isomorphic to an automorphism group of maximal possible order 84(g — 1) of a closed Riemann surface of genus g > 1. The Hurwitz groups are exactly the finite quotients, with torsionfree kernel, of the (2, 3, 7)-triangle group. Using equivariant Heegaard splittings, we consider a generalization for closed orientable hyperbolic 3-manifolds, leading to the concept of a maximally symmetric hyperbolic 3-manifold. We give an analogous purely algebraic characterization of the corresponding class of finite groups. We also discuss the following question: given a Hurwitz action on a closed Riemann surface, when does it extend to a compact hyperbolic 3-manifold with the given surface as totally geodesic boundary?
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