Geometry & Topology

Commensurations of the Johnson kernel

Tara E Brendle and Dan Margalit

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

Article information

Geom. Topol., Volume 8, Number 3 (2004), 1361-1384.

Received: 15 June 2004
Revised: 25 October 2004
Accepted: 25 October 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Secondary: 20F38: Other groups related to topology or analysis 20F36: Braid groups; Artin groups

Torelli group mapping class group Dehn twist


Brendle, Tara E; Margalit, Dan. Commensurations of the Johnson kernel. Geom. Topol. 8 (2004), no. 3, 1361--1384. doi:10.2140/gt.2004.8.1361.

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