Addendum to: Commensurations of the Johnson kernel

Abstract

Let $\mathcal{K}\left(S\right)$ be the subgroup of the extended mapping class group, $Mod\left(S\right)$, generated by Dehn twists about separating curves. In our earlier paper, we showed that $Comm\left(\mathcal{K}\left(S\right)\right)\cong Aut\left(\mathcal{K}\left(S\right)\right)\cong Mod\left(S\right)$ when $S$ is a closed, connected, orientable surface of genus $g\ge 4$. By modifying our original proof, we show that the same result holds for $g\ge 3$, thus confirming Farb’s conjecture in all cases (the statement is not true for $g\le 2$).