The word problem and the metric for the Thompson-Stein groups

TL;DR

This paper investigates the Thompson-Stein groups, providing solutions to the word problem, characterizing minimal tree-pair diagrams, and analyzing the metric's properties, especially highlighting differences when the number of parameters exceeds one.

Contribution

It introduces a unique normal form for elements of F(n_1,...,n_k), solves the word problem, and analyzes the metric, revealing non-quasi-isometric behavior for k>1.

Findings

Unique normal form for F(n_1,...,n_k) elements

Solution to the word problem for these groups

The metric is not quasi-isometric to tree size when k>1

Abstract

We consider the Thompson-Stein 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 tree-pair diagram representatives of elements may not be unique when k>1. We establish how to find minimal tree-pair diagram representatives of elements of F(n_1,...,n_k), and we prove several theorems describing the equivalence of trees and tree-pair 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 quasi-isometric to the number of leaves or caret in the minimal tree-pair diagram, as is the case when k=1.