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Property (T) in k-gonal random groups

Part of: Probabilistic methods in group theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  22 February 2022

MurphyKate Montee
MurphyKate Montee*
Affiliation:
Department of Mathematics and Statistics, Carleton College, Northfield, MN 55057, USA
*
E-mail: murphykate.montee@gmail.com
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Abstract

The k-gonal models of random groups are defined as the quotients of free groups on n generators by cyclically reduced words of length k. As k tends to infinity, this model approaches the Gromov density model. In this paper, we show that for any fixed $d_0 \in (0, 1)$ , if positive k-gonal random groups satisfy Property (T) with overwhelming probability for densities $d >d_0$ , then so do jk-gonal random groups, for any $j \in \mathbb{N}$ . In particular, this shows that for densities above 1/3, groups in 3k-gonal models satisfy Property (T) with probability 1 as n approaches infinity.

Keywords

random groupsproperty (T)

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20P05: Probabilistic methods in group theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 734 - 738
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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