Commensurations of the Johnson kernel

Abstract

Let K be the subgroup of the extended mapping class group, Mod(S), 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(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.html

References in corpus (1)

• K_g is not finitely generated

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• Commensurators of surface braid groups
• Automorphisms of the Torelli complex for the one-holed genus two surface
• The co-Hopfian property of surface braid groups
• Commensurations of $\mathrm{Aut}(F_N)$ and its Torelli subgroup
• Injective Simplicial Maps of the Arc Complex on Nonorientable Surfaces
• Finite rigid sets of the non-separating curve complex