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Random triangular groups at density $1/3$

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

Published online by Cambridge University Press:  27 November 2014

Sylwia Antoniuk ,
Tomasz Łuczak  and
Jacek Świa̧tkowski
Sylwia Antoniuk
Affiliation:
Adam Mickiewicz University, Faculty of Mathematics and Computer Science, ul. Umultowska 87, 61-614 Poznań, Poland email antoniuk@amu.edu.pl
Tomasz Łuczak
Affiliation:
Adam Mickiewicz University, Faculty of Mathematics and Computer Science, ul. Umultowska 87, 61-614 Poznań, Poland email tomasz@amu.edu.pl
Jacek Świa̧tkowski
Affiliation:
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland email swiatkow@math.uni.wroc.pl
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Abstract

Let ${\rm\Gamma}(n,p)$ denote the binomial model of a random triangular group. We show that there exist constants $c,C>0$ such that if $p\leqslant c/n^{2}$, then asymptotically almost surely (a.a.s.) ${\rm\Gamma}(n,p)$ is free, and if $p\geqslant C\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ has Kazhdan’s property (T). Furthermore, we show that there exist constants $C^{\prime },c^{\prime }>0$ such that if $C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ is neither free nor has Kazhdan’s property (T).

Keywords

random groupKazhdan’s propertyspectral gaprandom graph

MSC classification

Primary: 20P05: Probabilistic methods in group theory
Secondary: 20F05: Generators, relations, and presentations 05C80: Random graphs
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 1 , January 2015 , pp. 167 - 178
Copyright
© The Author(s) 2014 

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