Embedding mapping class groups into finite products of trees

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 ${\ell }^p$ compression exponent $1$ for all $p\ge 1$ and they quasi-isometrically embed into ${\ell }^1(\mathbb{N})$. 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.

29 pages, 19 figures, to appear in Groups Geom. Dyn

References in corpus (2)

• Embedding relatively hyperbolic groups in products of trees
• Assouad-Nagata dimension of tree-graded spaces

Cited by in corpus (6)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• On metric relative hyperbolicity
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• Extending higher dimensional quasi-cocycles
• Hyperplanes of Squier's cube complexes
• Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic