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Extending finite group actions on surfaces to hyperbolic 3-manifolds

Published online by Cambridge University Press:  24 October 2008

Monique Gradolato  and
Bruno Zimmermann
Monique Gradolato
Affiliation:
Università degli Studi di Trieste, Dipartimento di Scienze Matematiche 34 100 Trieste, Italy
Bruno Zimmermann
Affiliation:
Università degli Studi di Trieste, Dipartimento di Scienze Matematiche 34 100 Trieste, Italy
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Extract

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 Fgbounds 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].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 1 , January 1995 , pp. 137 - 151
Copyright
Copyright © Cambridge Philosophical Society 1995

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