Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

Authors: John Crisp and Bert Wiest
Journal: Trans. Amer. Math. Soc. 359 (2007), 5485-5503
MSC (2000): Primary 20F36, 05C25
Published electronically: June 22, 2007
MathSciNet review: 2327038
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Abstract: We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group $ \textsl{Diff}(D^2,\partial D^2,\textsl{vol})$ of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $ L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $ F_n$ and $ \mathbb{Z}^n$ for all $ n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the group $ \textsl{Diff}(D^2,\partial D^2,\textsl{vol})$. Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3-manifold.

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John Crisp
Affiliation: Institut de Mathémathiques de Bourgogne (IMB), UMR 5584 du CNRS, Université de Bourgogne, 9 avenue Alain Savary, B.P. 47870, 21078 Dijon cedex, France

Bert Wiest
Affiliation: IRMAR, UMR 6625 du CNRS, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes, France

Keywords: Hyperbolic group, right-angled Artin group, braid group
Received by editor(s): July 6, 2005
Received by editor(s) in revised form: October 4, 2005
Published electronically: June 22, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.