The word problem and the metric for the ThompsonStein groups
Abstract
We consider the ThompsonStein group F(n_1,...,n_k) for integers n_1,...,n_k and k greater than 1. We highlight several differences between the cases k=1$ and k>1, including the fact that minimal treepair diagram representatives of elements may not be unique when k>1. We establish how to find minimal treepair diagram representatives of elements of F(n_1,...,n_k), and we prove several theorems describing the equivalence of trees and treepair diagrams. We introduce a unique normal form for elements of F(n_1,...,n_k) (with respect to the standard infinite generating set developed by Melanie Stein) which provides a solution to the word problem, and we give sharp upper and lower bounds on the metric with respect to the standard finite generating set, showing that in the case k>1, the metric is not quasiisometric to the number of leaves or caret in the minimal treepair diagram, as is the case when k=1.
 Publication:

arXiv eprints
 Pub Date:
 November 2008
 arXiv:
 arXiv:0811.3036
 Bibcode:
 2008arXiv0811.3036W
 Keywords:

 Mathematics  Group Theory;
 20F65
 EPrint:
 v1: 33 pages, 14 figures v2: 23 pages, 12 figures, revised to improve readability and make arguments more concise