Automorphisms of Torelli groups

TL;DR

This paper proves that automorphisms of the Torelli group for closed orientable surfaces of genus at least 3 are all induced by surface diffeomorphisms, extending known results to a broader class of surfaces.

Contribution

It establishes a new fundamental link between Torelli group automorphisms and surface diffeomorphisms for genus ≥ 3, generalizing previous results for higher genus.

Findings

Automorphisms of Torelli groups are induced by surface diffeomorphisms.

The result applies to surfaces of genus at least 3.

Extension of known automorphism results to lower genus surfaces.

Abstract

In this paper, we prove that each automorphism of the Torelli group of a surface is induced by a diffeomorphism of the surface, provided that the surface is a closed, connected, orientable surface of genus at least 3. This result was previously announced by Benson Farb for genus at least 4 and has recently been announced by Benson Farb and Nikolai Ivanov for subgroups of finite index in the Torelli group for surfaces of genus at least 5. This result is also directly analogous to previous results established for classical braid groups, mapping class groups, and surface braid groups.