Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Bent walls for random groups in the square and hexagonal model
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by Tomasz Odrzygóźdź;
Trans. Amer. Math. Soc. 378 (2025), 3779-3822
DOI: https://doi.org/10.1090/tran/9141
Published electronically: March 19, 2025

Abstract:

We consider two random group models: the hexagonal model and the square model, defined as the quotient of a free group by a random set of reduced words of length four and six respectively. Our first main result is that in the hexagonal model there exists sharp density threshold for Kazhdan’s Property (T) and it equals $\frac {1}{3}$. Our second main result is that for densities $< \frac {3}{8}$ a random group in the square model with overwhelming probability does not have Property (T). Moreover, we provide a new version of the Isoperimetric Inequality that concerns non-planar diagrams and we introduce new geometrical tools to investigate random groups: trees of loops, diagrams collared by a tree of loops and specific codimension one structures in the Cayley complex, called bent hypergraphs.
References
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Bibliographic Information
  • Tomasz Odrzygóźdź
  • Affiliation: Institute of Mathematics, Polish Academy of Science, Warsaw, Śniadeckich 8, Poland
  • MR Author ID: 1133983
  • Email: tomaszo@impan.pl
  • Received by editor(s): May 3, 2020
  • Received by editor(s) in revised form: April 29, 2023, January 24, 2024, and January 29, 2024
  • Published electronically: March 19, 2025
  • Additional Notes: This paper was created as a result of the research project UMO-2016/21/N/ST1/02594 financed by the Polish National Science Center.
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 3779-3822
  • MSC (2020): Primary 20F65
  • DOI: https://doi.org/10.1090/tran/9141
  • MathSciNet review: 4907944