Soficity, short cycles and the Higman group
Group Theory
2017-11-02 v4 Number Theory
Abstract
This is a paper with two aims. First, we show that the map from to itself defined by exponentiation has few -cycles -- that is to say, the number of cycles of length three is . 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 to itself, locally like an exponential map, yet satisfying a recurrence property.
Cite
@article{arxiv.1512.02135,
title = {Soficity, short cycles and the Higman group},
author = {Harald A. Helfgott and Kate Juschenko},
journal= {arXiv preprint arXiv:1512.02135},
year = {2017}
}
Comments
27 pages. Revised version. We address results by Glebsky and Kassabov-Kuperberg-Riley that go against the heuristic, allowing for the construction of functions that behave like $x\to m^x$, $m\ne 2$ and satisfy a recurrence property