Convexity Properties of the Diestel-Leader Group Γ_3(2)

The Diestel-Leader groups are a family of groups first introduced in 2001 by Diestel and Leader in [7]. In this paper, we demonstrate that the Diestel-Leader group Γ3(2) is not almost convex with respect to a particular generating set S. Almost convexity is a geometric property that has been shown by Cannon [3] to guarantee a solvable word problem (that is, in any almost convex group there is a finite-step algorithm to determine if two strings of generators, or “words”, represent the same group element). Our proof relies on the word length formula given by Stein and Taback in [10], and we construct a family of group elements X that contradicts the almost convexity condition. We then go on to show that Γ3(2) is minimally almost convex with respect to S.