Nilprogressions and groups with moderate growth

Breuillard, Emmanuel; Tointon, Matthew

doi:10.1016/j.aim.2015.11.025

Cited by 46 publications

(60 citation statements)

References 44 publications

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“…Proof. We essentially reproduce the proof of the closely related [4,Lemma 2.7]. If S n+1 xH = S n xH for a given n then it follows by induction that S r xH = S n xH for every r ≥ n, and hence that G = S n xH.…”

Section: Uniform Measurement Of Index By Almost-invariant Measuresmentioning

confidence: 94%

Commuting probabilities of infinite groups

Tointon

2020

J. London Math. Soc.

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Let G be a group, and let M = (µn) ∞ n=1 be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to µn commute. Antolín, Martino and Ventura define the degree of commutativity dcM (G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if µn is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (µn) is a sequence of almost-invariant measures on G, we show that if dcM (G) > 5 8 then G is abelian; if dcM (G) ≥ 1 2 + ε then the centre of G has index at most ε −1 ; and if dcM (G) ≥ α > 0 then G contains a normal subgroup Γ of index at most ⌈α −1 ⌉ and a normal subgroup H of cardinality at most exp(O(α −O(1) )) such that H ⊂ Γ and Γ/H is abelian. We also describe some general conditions on (µn) under which such theorems hold. These results generalise results for finite groups due to Gustafson and P. M. Neumann, and generalise and quantify a result for certain residually finite groups of subexponential growth due to Antolín, Martino and Ventura. We also present an application to conjugacy ratios, showing that if the sequence of word-metric balls in a finitely generated group is both left-and right-Følner then the conjugacy ratio is equal to the degree of commutativity. Combined with our main results, this generalises a result of Ciobanu, Cox and Martino.

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Section: Uniform Measurement Of Index By Almost-invariant Measuresmentioning

confidence: 94%

Commuting probabilities of infinite groups

Tointon

2020

J. London Math. Soc.

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“…For all non-abelian finite simple groups, Breuillard and Tointon [8] also obtained a diameter bound of max(|G| , C ) for arbitrary > 0 and a constant C depending only on . The diameter bounds in all these previous results depend poorly on the rank of the group.…”

Section: History and Backgroundmentioning

confidence: 99%

A diameter bound for finite simple groups of large rank

Biswas

Yang²

2017

J. London Math. Soc.

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Given a non-abelian finite simple group G of Lie type, and an arbitrary symmetric generating set S, it is conjectured by László Babai that its Cayley graph Γ(G, S) will have a diameter bound of (log |G|) O(1) . However, little progress has been made when the rank of G is large. In this article, we shall show that if G has rank n, and its base field has bounded size, then the diameter of Γ(G, S) would be bounded by exp (O(n(log n) 3 )).

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“…The basic idea behind our proof of Theorem 1.11 is to control the growth of S m in terms of the growth of a certain nilprogression of bounded rank and step. In its simplest form, the tool that allows us to do this is the following result, which essentially appeared in [36] and was implicit in [13]. Remark.…”

Section: Sets Of Polynomial Growth In Terms Of Progressionsmentioning

confidence: 99%

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Tessera

Tointon²

2018

Discrete Anal

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We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of the Freiman-Bilu result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a Lie-algebra version of the geometry-of-numbers argument at the centre of that result. We also present some applications. We verify a conjecture of Benjamini that if S is a symmetric generating set for a group such that 1 ∈ S and |S n | ≤ Mn D at some sufficiently large scale n then S exhibits polynomial growth of the same degree D at all subsequent scales, in the sense that |S r | M,D r D for every r ≥ n. Our methods also provide an important ingredient in a forthcoming companion paper in which we reprove and sharpen a result about scaling limits of vertex-transitive graphs of polynomial growth due to Benjamini, Finucane and the first author. We also note that our arguments imply that every approximate group has a large subset with a large quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent group of bounded rank and step.

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Nilprogressions and groups with moderate growth

Cited by 46 publications

References 44 publications

Commuting probabilities of infinite groups

Commuting probabilities of infinite groups

A diameter bound for finite simple groups of large rank

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