Automorphisms of the Torelli complex for the one-holed genus two surface

TL;DR

This paper investigates the automorphisms and superinjective maps of the Torelli complex for a genus two surface with one boundary, revealing structural properties and subgroup behaviors.

Contribution

It characterizes automorphisms of the Torelli complex and describes isomorphisms between finite index subgroups, advancing understanding of the Torelli group's structure.

Findings

Automorphisms of the Torelli complex are fully described.

Finite index subgroups of the Torelli group are co-Hopfian.

Isomorphisms between finite index subgroups are characterized.

Abstract

Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for S into itself, and show that any finite index subgroup of the Torelli group for S is co-Hopfian.