Addendum to: Commensurations of the Johnson kernel

TL;DR

This paper extends previous results on the automorphism and commensurator groups of the Johnson kernel, confirming Farb's conjecture for surfaces of genus at least 3.

Contribution

It proves that for genus g ≥ 3, the automorphism and commensurator groups of the Johnson kernel are isomorphic to the extended mapping class group, confirming Farb's conjecture.

Findings

Automorphism group of K(S) is isomorphic to Mod(S) for g ≥ 3.

Commensurator group of K(S) is isomorphic to Mod(S) for g ≥ 3.

The result is not valid for genus less than 3.

Abstract

Let K(S) be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. In our earlier paper, we showed that Comm(K(S)) and Aut(K(S)) are both isomorphic to Mod(S) when S is a closed, connected, orientable surface of genus g at least 4. By modifying our original proof, we show that the same result holds for g at leat 3, thus confirming Farb's conjecture in all cases (the statement is not true for any g less than 3).