Volume 8, issue 3 (2004)

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Commensurations of the Johnson kernel

Tara E Brendle and Dan Margalit

Geometry & Topology 8 (2004) 1361–1384
 arXiv: math.GT/0404445
Abstract

Let $\mathsc{K}$ 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 $Comm\left(\mathsc{K}\right)\cong Aut\left(\mathsc{K}\right)\cong Mod\left(S\right)$. More generally, we show that any injection of a finite index subgroup of $\mathsc{K}$ into the Torelli group $\mathsc{ℐ}$ of $S$ is induced by a homeomorphism. In particular, this proves that $\mathsc{K}$ is co-Hopfian and is characteristic in $\mathsc{ℐ}$. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of $\mathsc{ℐ}$ into $\mathsc{ℐ}$ is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.

A correction was submitted 30 Apr 2018 and posted 11 Nov 2018 in an online supplement.
Keywords
Torelli group, mapping class group, Dehn twist
Mathematical Subject Classification 2000
Primary: 57S05
Secondary: 20F38, 20F36
Supplementary material

Correction (posted 11 Nov 2018)