Commensurations of the Johnson kernel

Abstract

Let $\mathcal{K}$ be the subgroup of the extended mapping class group, $Mod\left(S\right)$, generated by Dehn twists about separating curves. Assuming that $S$ is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that $Comm\left(\mathcal{K}\right)\cong Aut\left(\mathcal{K}\right)\cong Mod(S)$. More generally, we show that any injection of a finite index subgroup of $\mathcal{K}$ into the Torelli group $\mathcal{ℐ}$ of $S$ is induced by a homeomorphism. In particular, this proves that $\mathcal{K}$ is co-Hopfian and is characteristic in $\mathcal{ℐ}$. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of $\mathcal{ℐ}$ into $\mathcal{ℐ}$ is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.