Commensurations of the Johnson kernel

Brendle, Tara E; Margalit, Dan

doi:10.2140/gt.2004.8.1361

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

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.

A correction was submitted 30 Apr 2018 and posted 11 Nov 2018 in an online supplement.

Keywords

Torelli group, mapping class group, Dehn twist

Mathematical Subject Classification 2000

Primary: 57S05

Secondary: 20F38, 20F36

References

Supplementary material

Correction (posted 11 Nov 2018)

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

Corrected: 11 November 2018

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

Volume 8, issue 3 (2004)