Volume 8, issue 3 (2004)

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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

arXiv: math.GT/0404445


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.

Torelli group, mapping class group, Dehn twist
Mathematical Subject Classification 2000
Primary: 57S05
Secondary: 20F38, 20F36
Forward citations
Received: 15 June 2004
Revised: 25 October 2004
Accepted: 25 October 2004
Published: 25 October 2004
Proposed: Walter Neumann
Seconded: Shigeyuki Morita, Joan Birman
Tara E Brendle
Department of Mathematics
Cornell University
310 Malott Hall
New York 14853
Dan Margalit
Department of Mathematics
University of Utah
155 South 1440 East
Salt Lake City
Utah 84112