Roots in the mapping class groups

doi:10.1112/plms/pdn036

Proceedings of the London Mathematical Society Advance Access originally published online on September 4, 2008
Proceedings of the London Mathematical Society 2009 98(2):471-503; doi:10.1112/plms/pdn036

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Roots in the mapping class groups

Christian Bonatti and Luis Paris

Institut de Mathématiques de Bourgogne
UMR 5584 du CNRS
Université de Bourgogne
BP 47870
21078 Dijon cedex
France
lparis@u-bourgogne.fr

Received 31 January 2007. Revision received 15 May 2008.

The purpose of this paper is to study the roots in the mappingclass groups. Let ${Sigma}$ be a compact oriented surface, possibly withboundary, let $P$ be a finite set of punctures in the interiorof ${Sigma}$ , and let $M$ ( ${Sigma}$ , $P$ ) denote the mapping class group (relativeto the boundary) of ( ${Sigma}$ , $P$ ). We prove that if ${Sigma}$ is of genus 1 andhas nonempty boundary, then each f $isin$ $M$ ( ${Sigma}$ ) has at most one m-rootup to conjugation for all m $≥$ 1. We prove that, however, if ${Sigma}$ is of genus at least 2, then there exist f, g $isin$ $M$ ( ${Sigma}$ , $P$ ) such thatf² = g², f is not conjugate to g, and none of the conjugatesof f commutes with g. Afterwards, we focus our study on theroots of the pseudo-Anosov elements. We prove that if ${partial}$ ${Sigma}$ $!=$ ${emptyset}$ , theneach pseudo-Anosov element f $isin$ $M$ ( ${Sigma}$ , $P$ ) has at most one m-root forall m $≥$ 1, but if ${partial}$ ${Sigma}$ = ${emptyset}$ then there exist two pseudo-Anosov elementsf, g $isin$ $M$ ( ${Sigma}$ ) (explicitly constructed) such that f^m = g^m for somem $≥$ 2, f is not conjugate to g, and none of the conjugates off commutes with g. Finally, we show that if ${Gamma}$ is a pure subgroupof $M$ ( ${Sigma}$ , $P$ ) and f $isin$ ${Gamma}$ , then f has at most one m-root in ${Gamma}$ for allm $≥$ 1. Note that there are finite-index pure subgroups in $M$ ( ${Sigma}$ , $P$ ).

2000 Mathematics Subject Classification 57M99, (primary) 57N05,57R30 (secondary).

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