Caret Substitution and Embeddings of Thompson's Groups
by
José Burillo
Tufts University
Coauthors: Sean Cleary, Melanie Stein

Thompson's group F is the group of PL homeomorphisms of the unit interval, each with finitely many breakpoints which are all at dyadic rationals and with slopes which are powers of 2. Generalizations F(p) of F are similarly defined except that the breakpoints occur in Z[1/p] and the slopes are all powers of p.

Each F(p) is finitely generated with p generators. We study the word metric with respect to these finite generating sets. We develop an algebraic estimate of the word metric and show it to be quasi-isometric to the word metric with respect to the finite generating set. We develop a geometric estimate of the word metric, coming from the understanding of elements of F(p) as given by maps between rooted trees of valence p+1 at each vertex, and show this also to be quasi-isometric to the word metric.

We use these algebraic and geometric estimates of the word metric to show that large families of embeddings of F(p) into F(q) for arbitrary p and q are quasi-isometric embeddings. These embeddings can be understood geometrically through the process of caret substitution on the reduced tree diagrams representing group elements of F(p)

http://www.tufts.edu/~jburillo

Date received: March 5, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-60.