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Mapping Class Groups of Hyperbolic Surfaces and Automorphism Groups of Graphs

Published online by Cambridge University Press:  04 December 2007

Paul Schmutz Schaller
Paul Schmutz Schaller
Affiliation:
Section de mathématiques, Université de Genève, CP 240, CH-1211 Geneva 24, Switzerland. E-mail: [email protected]
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Abstract

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Let M be a hyperbolic surface and Γ(M) its extended mapping class group. We show that Γ(M) is isomorphic to the automorphism group of the following graph G(M). The set of vertices of G(M) is the set S(M) of nonseparating simple closed geodesics of M. Two vertices u and v of S(M) are related by an edge if u and v intersect exactly once in M. The graph G(M) can be thought of as a combinatorial model for M.

Keywords

mapping class grouphyperbolic surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 122 , Issue 3 , July 2000 , pp. 243 - 260
Copyright
© 2000 Kluwer Academic Publishers