Automorphisms of the Torelli Complex for the One-holed Genus Two Surface

Abstract

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