Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Soficity, short cycles, and the Higman group


Authors: Harald A. Helfgott and Kate Juschenko
Journal: Trans. Amer. Math. Soc. 371 (2019), 2771-2795
MSC (2010): Primary 05A05; Secondary 43A07
DOI: https://doi.org/10.1090/tran/7534
Published electronically: November 5, 2018
MathSciNet review: 3896097
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ 3$ is $ o(p)$. This improves on previous bounds.

Our second objective is to contribute to an ongoing discussion on how to find a nonsofic 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 05A05, 43A07

Retrieve articles in all journals with MSC (2010): 05A05, 43A07


Additional Information

Harald A. Helfgott
Affiliation: Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany; and IMJ-PRG, UMR 7586, 58 avenue de France, Bâtiment Sophie Germain, Case 7012, 75013 Paris CEDEX 13, France
Email: helfgott@math.univ-paris-diderot.fr

Kate Juschenko
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Email: kate.juschenko@gmail.com

DOI: https://doi.org/10.1090/tran/7534
Received by editor(s): April 4, 2017
Received by editor(s) in revised form: October 9, 2017, October 23, 2017, and January 8, 2018
Published electronically: November 5, 2018
Additional Notes: The first author is currently supported by ERC Consolidator grant 648329 (GRANT) and by funds from his Humboldt Professorship. Part of the work toward this paper was carried out while he visited the Chebyshev Laboratory (St. Petersburg, Russia) and IMPA (Rio de Janeiro, Brazil).
Part of the work toward this paper was carried out while the second author visited the Weizmann Institute (Rehovot, Israel) and the Bernoulli Centre (Lausanne, Switzerland).
Article copyright: © Copyright 2018 American Mathematical Society