Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Extending finite group actions on surfaces to hyperbolic 3-manifolds

Monique Gradolatoa1 and Bruno Zimmermanna1

a1 Università degli Studi di Trieste, Dipartimento di Scienze Matematiche 34 100 Trieste, Italy

Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold Fg of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed Riemann surface). We say that the G-action on Fg bounds a hyperbolic 3-manifold M if M is a compact orientable hyperbolic 3-manifold with totally geodesic boundary Fg (as the only boundary component) such that the G-action on Fg extends to a G-action on M by isometrics. Symmetrically we will also say that the 3-manifold M bounds the given G-action. We are especially interested in Hurwitz actions, i.e. finite group actions on surfaces of maximal possible order 84(g — 1); the corresponding finite groups are called Hurwitz groups. First examples of bounding and non-bounding Hurwitz actions were given in [16].

(Received November 30 1993)

(Revised March 23 1994)