The curve complex as a coset intersection complex

TL;DR

This paper demonstrates that a certain coset intersection complex related to the mapping class group of a surface is quasi-isometric and homotopy equivalent to the curve complex, with their combinatorial structures being closely related.

Contribution

It introduces a new coset intersection complex that models the curve complex and establishes its geometric and combinatorial equivalence to the curve complex.

Findings

The coset intersection complex is quasi-isometric to the curve complex.

The complexes are homotopy equivalent.

The automorphism group of the coset intersection complex is the extended mapping class group.

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.