The curve complex as a coset intersection complex

Abstract

We show that there is a collection of subgroups of the mapping class group of a surface such that the associated coset intersection complex is quasi-isometric and homotopy equivalent to the curve complex. Moreover, we prove that these two complexes are combinatorially equivalent in the sense that one can be obtained from the other via taking a nerve. As an application, we prove that the automorphism group of this coset intersection complex is the extended mapping class group, a result in the spirit of for Ivanov's metaconjecture.