Random groups and property (<i>T</i>
): Żuk's theorem revisited

Kotowski, Marcin; Kotowski, Michał

doi:10.1112/jlms/jdt024

Cited by 39 publications

(62 citation statements)

References 13 publications

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“…It should be mentioned that, as is well known in the theory of random structures, the asymptotic behavior of Γ(n, t) and Γ(n, p) can be proved to be very similar provided t ∼ Np. We remark that Γ(n, t) is basically identical with the model of random group introduced byŻuk [9] who also pointed out the connection between his model and a better known construction of random groups introduced by Gromov [4] (a more precise explanation of this relationship is given in [6,7]).…”

mentioning

confidence: 60%

Collapse of random triangular groups: a closer look

Antoniuk

Łuczak

Świątkowski

2014

Bulletin of the London Mathematical Society

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Abstract. The random triangular group Γ(n, t) is a group given by a presentation P = S|R , where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of Γ(n, t) is in some respects similar to that of widely studied density random group introduced by Gromov. In particular, it is known that if t ≤ n 3/2−ε for some ε > 0, then with probability 1 − o(1) Γ(n, t) is infinite and hyperbolic, while for t ≥ n 3/2+ε , with probability 1 − o(1) it is trivial. In this note we show that Γ(n, t) collapses provided only that t ≥ Cn 3/2 for some constant C > 0.By S|R we denote a presentation P of a group, where S is the set of generators and R the set of relations. A presentation P is a triangular presentation if the set R of relations consists of distinct cyclically reduced words of length three only (over the alphabet S ∪S −1 which consists of generators and their formal inverses). A triangular group is a group given by a triangular presentation.In the paper we investigate the properties of groups given by presentations P = S|R , where S is a set of n generators and R is a random subset of the set T of all N = 2n(4n 2 − 6n + 3) ∼ 8n 3 distinct cyclically reduced words of length three, that is words of the form abc, where a = b −1 , b = c −1 and c = a −1 . Here we consider two models of a random triangular group. In the random uniform model Γ(n, t), the set of relations R is chosen uniformly at random from the family of all N t subsets of T of size t. In the binomial model Γ(n, p), we select each element from T independently with probability p. We shall study the asymptotic properties of these two models as n → ∞. In particular, we say that for a given function t(n) [or p(n)] the group Γ(n, t(n)) [resp. Γ(n, p(n))] has a property a.a.s. (asymptotically almost surely) if the probability that the random group has this property tends to 1 as n → ∞. It should be mentioned that, as is well known in the theory of random structures, the asymptotic behavior of Γ(n, t) and Γ(n, p) can

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confidence: 60%

Collapse of random triangular groups: a closer look

Antoniuk

Łuczak

Świątkowski

2014

Bulletin of the London Mathematical Society

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“…However, he did not fully justified this statement. Recently, Kotowski and Kotowski [5] showed how to modifyŻuk's argument for the permutation model to make it work for the triangular model Γ(n, t) as well, completing the proof of the following theorem.…”

Section: Then γ Has Kazhdan's Property (T)mentioning

confidence: 88%

“…(asymptotically almost surely) if the probability of Γ(n, p) having this property tends to 1 as n → ∞. FromŻuk's result [10] (see also [5]) it follows that for every ǫ > 0 a.a.s. Γ(n, p) is free provided p ≤ n −2−ǫ , and it has a.a.s.…”

Section: Introductionmentioning

confidence: 99%

Random triangular groups at density

2014

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“…The most amusing is Zuk's method which enables (sometimes) to deduce property (T ) from a presentation of Γ by generators and relations. For example it shows property (T ) for some random groups (see also [KK13]). This is very different than the way Kazhdan produced the first groups with property (T ) and it shows that property (T ) is not such a rare property.…”

Section: High Dimensional Expanders: Spectral Gapmentioning

confidence: 98%

High Dimensional Expanders

Lubotzky

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways (cf.[Lub94], [HLW06], [Lub12] and the references therein). In the last decade, a theory of "high dimensional expanders" has begun to emerge. The goal of the current paper is to describe some paths of this new area of study.

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Random groups and property (T ): Żuk's theorem revisited

Cited by 39 publications

References 13 publications

Collapse of random triangular groups: a closer look

Collapse of random triangular groups: a closer look

Random triangular groups at density

High Dimensional Expanders

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