The square model for random groups

Odrzygóźdź, Tomasz

doi:10.4064/cm142-2-5

Cited by 9 publications

(13 citation statements)

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“…[Oll04]). This argument also transfers to the k-angular model [ARD20]: see [Odr16] for the case of k = 4, as well as a generalisation of the argument to a wider class of diagrams. In the opposite direction to Property (T), there are many results known about the lack of Property (T) in various models of random groups.…”

Section: Property (T) In Random Groupsmentioning

confidence: 80%

“…If we keep n fixed and let k tend to infinity, then we obtain the Gromov density model, as introduced in [Gro93], whereas if we fix k and let n tend to infinity we obtain the k-angular model, as introduced in [ARD20]. The kangular model was first studied for k = 3 (the triangular model ) by Żuk in [Ż03] and for k = 4 (the square model ) by Odrzygóźdź in [Odr16].…”

Section: Property (T) In Random Groupsmentioning

confidence: 99%

“…As observed in [Odr19] this implies that for any k ≥ 3, a random group in the k-angular model at density d < 5/24 does not have Property (T). Groups in the triangular model are free at densities less than 1/3 [ALuS15], groups in the square model are free at densities less than 1/4 [Odr16], and groups in the k-angular model are free for d < 1/k [ARD20]. Furthermore, groups in the square model are virtually special for d < 1/3 [Duo17, Odr18] and contain a codimension-1 subgroup for d < 3/8 [Odr19].…”

Section: Property (T) In Random Groupsmentioning

confidence: 99%

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Property (T) in density-type models of random groups

Ashcroft¹

2021

Preprint

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We study Property (T) in the Γ(n, k, d) model of random groups: as k tends to infinity this gives the Gromov density model, introduced in [Gro93]. We provide bounds for Property (T) in the k-angular model of random groups, i.e. the Γ(n, k, d) model where k is fixed and n tends to infinity. We also prove that for d > 1/3, a random group in the Γ(n, k, d) model has Property (T) with probability tending to 1 as k tends to infinity, strengthening the results of Żuk and Kotowski-Kotowski, who consider only groups in the Γ(n, 3k, d) model.

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Section: Property (T) In Random Groupsmentioning

confidence: 80%

Section: Property (T) In Random Groupsmentioning

confidence: 99%

Section: Property (T) In Random Groupsmentioning

confidence: 99%

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Property (T) in density-type models of random groups

Ashcroft¹

2021

Preprint

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“…In [Odr16] the author introduced the square model for random group and prove that such random groups for densities < 1 3 , with overwhelming probability, do not satisfy Property (T), and that for densities < 1 2 are infinite and hyperbolic. In his next paper [Odr18] the author proved that for densities < 3 10 random groups in the square model act properly and cocompactly on a CAT(0) cube complex.…”

Section: Introductionmentioning

confidence: 99%

Bent walls for random groups in the square and hexagonal model

Odrzygóźdź¹

2019

Preprint

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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 this model there exists sharp density threshold for Kazhdan's Property (T) and it equals 1 3 . Our second main result is that for densities < 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.This paper was created as a result of the research project UMO-2015/18/M/ST1/00050 financed by the Polish National Science Center.

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“…In [Odr16] we introduced the square model for random groups, where we draw at random relations of length four. The motivation was that the Cayley complex of such a group has a natural structure of a square complex, which should be easier to analyze than the polygonal Cayley complexes for groups in the Gromov model.…”

Section: Introductionmentioning

confidence: 99%

Cubulating random groups in the square model

Odrzygóźdź

2018

Isr. J. Math.

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Our main result is that for densities < 3 10 a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality, to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure of a space with walls on the Cayley complex of a random group. Then we show that the natural action of a random group on this space with walls is proper, which gives the proper action of a random group on a CAT(0) cube complex. 1 2 TOMASZ ODRZYGÓŹDŹ Proof. The residual finiteness results from [Ago13, Corollary 1.2] and the Haagerup Property by a folklore remark (see for example [CMV04]).To obtain Theorem 1.2 we needed to generalize the isoperimetric inequality ([Oll07, Theorem 2]) to a class of non-planar diagrams (see Theorem 2.5). This generalization has already proved to be useful in other work on random groups (see for example [MP15]). Theorem 2.5 is stated in the regime of the square model, but the reasoning can be easily repeated in the Gromov model (see [Odr14] for some version of it). The main idea of the proof of Theorem 1.2 is based on introducing modified hypergraps, which provide a structure of a space with walls on the Cayley complex of a random group, that has desired metric properties. This modified walls allowed us to construct a proper action of a random group on a space with walls, which was impossible to obtain using standard hypergraphs (introduced in [OW11] for the Gromov model and in [Odr16] for the square model).Organization. The paper is organized as follows: first we prove a generalization of the isoperimetric inequality, then using this we introduce "corrected" hypergraphs that provide the structure of a space with walls on the Cayley complex of a random group. Finally we show that a random group acts properly on this space with walls. The idea of correcting hypergraphs was inspired by [MP15].

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The square model for random groups

Cited by 9 publications

References 4 publications

Property (T) in density-type models of random groups

Property (T) in density-type models of random groups

Bent walls for random groups in the square and hexagonal model

Cubulating random groups in the square model

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