Automorphisms of the Torelli complex and the complex of separating curves

TL;DR

This paper determines the automorphism groups of the Torelli complex and the complex of separating curves for most surfaces, revealing their deep connection to the extended mapping class group and advancing understanding of surface symmetries.

Contribution

It computes the automorphism groups of key complexes related to surface topology for nearly all surfaces, establishing their isomorphism with the extended mapping class group.

Findings

Automorphism groups of the complexes are identified for all but finitely many surfaces.

The abstract commensurators of the Torelli group and Johnson kernel are shown to be isomorphic to the extended mapping class group.

Provides new insights into the symmetry structures of surface-related complexes.

Abstract

We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping class group.