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

We consider the Thompson-Stein group F (n1, ..., nk) where n1, ..., nk ∈ {2, 3, 4, ...}, k ∈ N. 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 (n1, ..., nk), and we prove several theorems describing the equivalence of trees and tree-pair diagrams. We introduce a unique normal form for elements of F (n1, ..., nk) (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.