Tame combing and almost convexity conditions

TL;DR

This paper constructs examples of groups, including Thompson's group F and certain Baumslag-Solitar groups, that have tame combings with linear radial tameness functions but are not minimally almost convex, expanding understanding of their Cayley complexes.

Contribution

It provides the first examples of such groups with these properties and develops new combinatorial descriptions of their Cayley complexes.

Findings

Constructed tame combings for Thompson's group F and BS(1,p) with p ≥ 3.

Expanded understanding of the Cayley complex of Thompson's group F.

Classified arrangements of 2-cells adjacent to edges in the Cayley complex.

Abstract

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the Baumslag-Solitar groups BS(1, p) with p \ge 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.