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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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References

REFERENCES

[1]Beardon, A. F.. The Geometry of Discrete Groups. (Springer-Verlag, 1983).CrossRefGoogle Scholar
[2]Conder, M.. Groups of minimal genus including C 2 extensions of PSL(2, q) for certain q. Quart. J. Math. Oxford (2) 38 (1987), 449–460.CrossRefGoogle Scholar
[3]Conder, M.. Maximal automorphism groups of symmetric Riemann surfaces with small genus. J. Algebra 114 (1988), 16–28.CrossRefGoogle Scholar
[4]Conder, M.. Hurwitz groups: a brief survey. Bull. Amer. Math. Soc. 23 (2) (1990), 359–370.CrossRefGoogle Scholar
[5]Coxeter, H. S. and Moser, W. O.. Generators and Relations for Discrete Groups (Springer-Verlag, 1957).CrossRefGoogle Scholar
[6]Dickson, L. E.. Linear Groups With an Exposition of the Galois Field Theory. (Dover Publications, 1958).Google Scholar
[7]Glover, H. and Sjerve, D.. Representing PSl 2(q) on a Riemann surface of least genus. L'Ens. Math. 31 (1985), 305–325.Google Scholar
[8]Glover, H. and Sjerve, D.. The genus of PSl 2(q). J. reine angew. Math. 380 (1987), 59–86.Google Scholar
[9]Macbeath, A. M.. Generators of the linear fractional groups. Proc. Symp. Pure Math. Number Theory, vol. XII (1968), 14–32.Google Scholar
[10]Meyerhoff, R.. A lower bound for the volume of hyperbolic 3-orbifolds. Duke Math. J. 57 (1988), 185–203.CrossRefGoogle Scholar
[11]Rosen, K. H.. Elementary Number Theory and its Applications. (Addison-Wesley, 1988).Google Scholar
[12]Sah, C. H.. Groups related to compact Riemann surfaces. Acta Math. 123 (1969), 13–42.CrossRefGoogle Scholar
[13]Suzuki, M.. Group Theory. Vol. I (Springer-Verlag, 1982).Google Scholar
[14]Thurston, W. P.. The Geometry and Topology of 3-manifolds. (Princeton University Lecture Notes, 1978–1979).Google Scholar
[15]Zimmermann, B.. Finite group actions on handlebodies and equivariant Heegaard genus for 3-manifolds. Topology Appl. 43 (1992), 263–274.CrossRefGoogle Scholar
[16]Zimmermann, B.. Hurwitz groups and finite group actions on hyperbolic 3-manifolds. To appear in J. London Math. Soc.Google Scholar