Soficity and variations on Higman's group

TL;DR

This paper explores the soficity of Higman's group and its variations, providing conditions under which these groups are sofic and analyzing the implications for the existence of non-sofic groups.

Contribution

It introduces new variations of Higman's group with criteria for soficity and examines the implications for pathological permutation behaviors.

Findings

Certain group variations are proven to be sofic under specific conditions.

Existence of permutations with unusual properties is demonstrated.

Results challenge previous suggestions linking soficity to non-sofic group existence.

Abstract

A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag--Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways. We construct variations on Higman's group replacing the Baumslag--Solitar group by other groups $G$. We give an elementary condition on $G$, enjoyed for example by $\mathbb{Z} \wr \mathbb{Z}$ and the integral Heisenberg group, under which the resulting group is sofic. We then use soficity to deduce that there exist permutations of $\mathbb{Z} / n\mathbb{Z}$ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.