Thompson’s group is distorted in the Thompson–Stein groups

Claire Wladis

Vol. 250 (2011), No. 2, 473–485

DOI: 10.2140/pjm.2011.250.473

Abstract

We show that the inclusion map of the generalized Thompson groups F(n_i) is exponentially distorted in the Thompson–Stein groups F(n₁,…,n_k) for k > 1. One consequence is that F is exponentially distorted in F(n₁,…,n_k) for k > 1 whenever n_i = 2^m for some m (whenever no i,m exist such that n_i = 2^m, there is no obviously “natural” inclusion map of F into F(n₁,…,n_k)). This is the first known example in which the natural embedding of one of the Thompson-type groups into another is not quasi-isometric.

Keywords

Thompson’s group, piecewise linear homeomorphism, Stein group, Higman group, quasi-isometrically embedded subgroup, distorted subgroup

Mathematical Subject Classification 2000

Primary: 20F65

Milestones

Received: 7 February 2010

Accepted: 17 May 2010

Published: 31 March 2011