Soficity, short cycles and the Higman group

TL;DR

This paper investigates the scarcity of 3-cycles in exponential maps over finite cyclic groups and explores implications for the soficity of the Higman group, advancing understanding of group approximations and cycle structures.

Contribution

It improves bounds on the number of 3-cycles in exponential maps and links these properties to the potential non-soficity of the Higman group.

Findings

Number of 3-cycles in exponential maps is o(p)

If Higman group is sofic, certain exponential-like maps with recurrence properties must exist

Provides new bounds and conditions related to sofic groups

Abstract

This is a paper with two aims. First, we show that the map from $\mathbb{Z}/p\mathbb{Z}$ to itself defined by exponentiation $x\to m^x$ has few $3$-cycles -- that is to say, the number of cycles of length three is $o(p)$. This improves on previous bounds. Our second objective is to contribute to an ongoing discussion on how to find a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from $\mathbb{Z}/p\mathbb{Z}$ to itself, locally like an exponential map, yet satisfying a recurrence property.