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Licensed Unlicensed Requires Authentication Published by De Gruyter October 30, 2018

The free group on n generators modulo n + u random relations as n goes to infinity

  • Yuan Liu EMAIL logo and Melanie Matchett Wood ORCID logo

Abstract

We show that, as n goes to infinity, the free group on n generators, modulo n+u random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen–Lenstra heuristics. For each n, these random groups belong to the few relator model in the Gromov model of random groups.

Award Identifier / Grant number: DMS-1652116

Award Identifier / Grant number: DMS-1301690

Funding statement: This work was done with the support of an American Institute of Mathematics Five-Year Fellowship, a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, a Vilas Early Career Investigator Award, and National Science Foundation grants DMS-1652116 and DMS-1301690.

Acknowledgements

We thank Nigel Boston, Persi Diaconis, Tullia Dymarz, Benson Farb, Turbo Ho, Peter Sarnak, Mark Shusterman, and Tianyi Zheng for helpful conversations. We thank Goulnara Arjantseva and the anonymous referee for feedback on an earlier version of this manuscript.

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Received: 2018-01-12
Revised: 2018-08-16
Published Online: 2018-10-30
Published in Print: 2020-05-01

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