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We consider the Thompson–Stein group  F ( n  1 ,…, n   k  ), where  n  1 , …,  n   k  ∈{2, 3, 4, …} and  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 ( 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 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.

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