Embedding mapping class groups into a finite product of trees

作者：Hume David^*

来源：Groups, Geometry, and Dynamics, 2017, 11(2): 613-647.

DOI：10.4171/GGD/410

• Tree-graded space
• quasi-tree
• embeddings
• mapping class group
• curve complex

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 de fined 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 l(p) compression exponent 1 for all p >= 1 and they quasi-isometrically embed into l(1)(IN). 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.

• 出版日期2017