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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060

Commensurations of the Johnson kernel

Tara E Brendle and Dan Margalit

Geometry & Topology 8 (2004) 1361–1384

DOI: 10.2140/gt.2004.8.1361

arXiv: math.GT/0404445

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)≈Mod(S). 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

Mathematical Subject Classification

Primary: 57S05

Secondary: 20F36, 20F38

References
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Publication

Received: 15 June 2004
Revised: 25 October 2004
Accepted: 25 October 2004
Published: 25 October 2004
Proposed: Walter Neumann
Seconded: Shigeyuki Morita, Joan Birman

Authors
Tara E Brendle
Department of Mathematics
Cornell University
310 Malott Hall
Ithaca
New York 14853
USA
Dan Margalit
Department of Mathematics
University of Utah
155 South 1440 East
Salt Lake City
Utah 84112
USA