Mapping Class Groups of Hyperbolic Surfaces and Automorphism Groups of Graphs

Abstract

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.