Embedding mapping class groups into a finite product 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.