Degree of commutativity of infinite groups

Antolín, Yago; Martino, Armando; Ventura, Enric

doi:10.1090/proc/13231

Cited by 20 publications

(51 citation statements)

References 10 publications

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“…Burillo and Ventura [5,Proposition 2.4] show that if S is a finite symmetric generating set for G containing the identity and satisfying (1.1), and M = (µ n ) ∞ n=1 is the sequence of probability measures on G defined by setting µ n to be the uniform probability measure on S n , then µ n (xH) → 1/[G : H] for every subgroup H of G and every x ∈ G (here, and elsewhere, for notational convenience we take 1/[G : H] to be zero when H is an infinite-index subgroup of G). This was in turn used in Antolín, Martino and Ventura's proof of Theorem 1.4 [1]. In the present paper we show that this convergence is uniform over H and x, which implies in particular that M detects index uniformly.…”

Section: Introductionsupporting

confidence: 68%

“…. such that dc µn i (G) ≥ α − o (1). Writing E (n) for expectation with respect to µ n , this means precisely that…”

Section: A Weak Neumann-type Theoremmentioning

confidence: 99%

“…Antolín, Martino and Ventura [1] approach this issue by considering sequences of finitely supported probability measures whose supports converge to the whole of G. Given a probability measure µ on G, define the degree of commutativity dc µ (G) of G with respect to µ via dc µ (G) = (µ × µ)({(x, y) ∈ G × G : xy = yx}).…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately we do not prove Conjecture 1.3 for the seqence of uniform probability measures on balls with respect to a fixed generating set that does not satisfy (1.1). As is noted in [1], in order to prove such a result it would be sufficient to prove that if a finite generating set S for a group G did not satisfy (1.1) then dc S (G) = 0.…”

Section: Introductionmentioning

confidence: 99%

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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: Introductionsupporting

confidence: 68%

“…. such that dc µn i (G) ≥ α − o (1). Writing E (n) for expectation with respect to µ n , this means precisely that…”

Section: A Weak Neumann-type Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Commuting probabilities of infinite groups

Tointon

2020

J. London Math. Soc.

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“…The degree of commutativity of a group has received a lot of attention recently, as its definition was extended to finitely generated infinite groups in [AMV17] to be dc X (G) = lim sup n→∞ |{(x, y) ∈ B G,X (n) 2 : ab = ba}| |B G,X (n)| 2 .…”

Section: Introductionmentioning

confidence: 99%

The Conjugacy Ratio of Groups

Cox

Ciobanu²,

Martino

2019

Proceedings of the Edinburgh Mathematical Society

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In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.

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Word problems for finite nilpotent groups

Camina¹,

Iñiguez²,

Thillaisundaram

2020

Arch. Math.

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Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that Nw(1) ≥ |G| k−1 , where for g ∈ G, the quantity Nw(g) is the number of k-tuples (g1,. .. , g k) ∈ G (k) such that w(g1,. .. , g k) = g. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, which states that Nw(g) ≥ |G| k−1 for g a w-value in G, and prove that Nw(g) ≥ |G| k−2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

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Degree of commutativity of infinite groups

Cited by 20 publications

References 10 publications

Commuting probabilities of infinite groups

Commuting probabilities of infinite groups

The Conjugacy Ratio of Groups

Word problems for finite nilpotent groups

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