Kazhdan groups have cost 1

Hutchcroft, Tom; Pete, Gábor

doi:10.1007/s00222-020-00967-6

Cited by 3 publications

(1 citation statement)

References 36 publications

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“…In [4], it was shown that any group with property (T) has cost 1. This raises the question if a similar theorem can be proved in the combinatorial context.…”

Section: Definition 14 ([1]mentioning

confidence: 99%

Cycles in graphs with geometric property (T)

Winkel¹

2022

Preprint

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We show that a sequence of graphs with uniformly bounded vertex degrees, number of vertices going to infinity, and with geometric property (T) has many small cycles. We also show that when a small part of such a sequence of graphs with geometric property (T) is changed, it still has geometric property (T), provided that it is still an expander. We use this to give an example of a sequence of graphs with geometric property (T) that has large cycle-free balls.

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“…In [4], it was shown that any group with property (T) has cost 1. This raises the question if a similar theorem can be proved in the combinatorial context.…”

Section: Definition 14 ([1]mentioning

confidence: 99%

Cycles in graphs with geometric property (T)

Winkel¹

2022

Preprint

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Cost of inner amenable groupoids

Tucker-Drob

Wróbel

2021

Proc. Amer. Math. Soc.

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Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have “fixed priced 1” in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that inner amenable groups have fixed price 1.

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The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

Pete

Timár

2022

Ann. Probab.

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We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely.This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres [LP16] in the negative.A version of our argument gives an example of a non-unimodular transitive graph where WUSF = FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang [Tan19].

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Kazhdan groups have cost 1

Abstract: We prove that every countably infinite group with Kazhdan's property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.

Cited by 3 publications

References 36 publications

Cycles in graphs with geometric property (T)

Cycles in graphs with geometric property (T)

Cost of inner amenable groupoids

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

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