Geometry & Topology, Vol. 8 (2004) Paper no. 37, pages 1361--1384.

Commensurations of the Johnson kernel

Tara E Brendle, Dan Margalit

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.

Keywords. Torelli group, mapping class group, Dehn twist

AMS subject classification. Primary: 57S05. Secondary: 20F38, 20F36.

DOI: 10.2140/gt.2004.8.1361

E-print: arXiv:math.GT/0404445

Submitted to GT on 15 June 2004. (Revised 25 October 2004.) Paper accepted 25 October 2004. Paper published 25 October 2004.

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Tara E Brendle, Dan Margalit
Department of Mathematics, Cornell University
310 Malott Hall, Ithaca, NY 14853, USA
Department of Mathematics, University of Utah
155 S 1440 East, Salt Lake City, UT 84112, USA


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