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.
Notes on file formats
Tara E Brendle, Dan Margalit
Department of Mathematics, Cornell University
310 Malott Hall,
Ithaca, NY 14853, USA
and
Department of Mathematics, University
of Utah
155 S 1440 East, Salt Lake City, UT 84112, USA
Email: brendle@math.cornell.edu, margalit@math.utah.edu
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