JournalsggdVol. 11, No. 2pp. 613–647

Embedding mapping class groups into a finite product of trees

  • David Hume

    UC Louvain, Louvain-La-Neuve, Belgium
Embedding mapping class groups into a finite product of trees cover
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Abstract

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad–Nagata dimension, direct embeddings exhibiting p\ell^p compression exponent 11 for all p1p\geq 1 and they quasi-isometrically embed into 1(N)\ell^1(\N). We deduce similar consequences for relatively hyperbolic groups whose parabolic subgroups satisfy such conditions.

In obtaining these results we also demonstrate that curve complexes of compact surfaces and coned-off graphs of relatively hyperbolic groups admit quasi-isometric embeddings into finite products of trees.

Cite this article

David Hume, Embedding mapping class groups into a finite product of trees. Groups Geom. Dyn. 11 (2017), no. 2, pp. 613–647

DOI 10.4171/GGD/410