Tame Combing and Almost Convexity Conditions

We explore relationships between the family of successively weaker almost convexity conditions, and successively weaker tame combing conditions. We show that both Thompson’s group F and the Baumslag-Solitar groups BS(1, p) with p ≥ 3 admit a tame combing with a linear radial tameness function. By earlier work of Belk and Bux showing that F is not minimally almost convex, and of Elder and Hermiller showing that BS(1, p) with p ≥ 7 is not minimally almost convex, in each case with respect to a standard generating set, this result provides examples of groups and generating sets satisfying a strong tame combing condition yet not even the weakest almost convexity condition on the same generating set. We also show that an inclusion on strong tame combing conditions is strict by showing that although BS(1, p) with the standard generators and p ≥ 8 does admit a tame combing with a linear radial tameness function, the linear coefficient of the tameness function must be greater than 1.