Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

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 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 $\Z^n$ 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 diffeomorphism group of the disk. 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, Bert Wiest. Quasi-isometrically embedded subgroups of braid and diffeomorphism groups. Transactions of the American Mathematical Society, American Mathematical Society, 2007, 11 (5485), pp.5503. ⟨hal-00005452⟩

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