Some orbits of free words that are determined by measures on finite groups

Hanany, Liam; Meiri, Chen; Puder, Doron

doi:10.1016/j.jalgebra.2020.03.013

Cited by 10 publications

(7 citation statements)

References 13 publications

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“…In the case of the simple commutator [x, y], this strengthens Theorem 1.4, as it only relies on measures on finite groups. On the other hand, unlike the current paper where we use specific families of groups (U N and generalized permutation groups), the finite groups relied upon in [HMP19] are not explicit.…”

Section: Do Word Measures Determine the Word?mentioning

confidence: 96%

Surface Words are Determined by Word Measures on Groups

Magee¹,

Puder²

2019

Preprint

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Every word w in a free group naturally induces a probability measure on every compact group G. For example, if w = [x, y] is the commutator word, a random element sampled by the w-measure is given by the commutator [g, h] of two independent, Haar-random elements of G. Back in 1896, Frobenius showed that if G is a finite group and ψ an irreducible character, then the expected value of ψ ([g, h]) is 1 ψ(e) . This is true for any compact group, and completely determines the [x, y]-measure on these groups. An analogous result holds with the commutator word replaced by any surface word.We prove a converse to this theorem: if w induces the same measure as [x, y] on every compact group, then, up to an automorphism of the free group, w is equal to [x, y]. The same holds when [x, y] is replaced by any surface word.The proof relies on the analysis of word measures on unitary groups and on orthogonal groups, which appears in separate papers, and on new analysis of word measures on generalized symmetric groups that we develop here.

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Section: Do Word Measures Determine the Word?mentioning

confidence: 96%

Surface Words are Determined by Word Measures on Groups

Magee¹,

Puder²

2019

Preprint

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“…..,p k by the additional relation x p 1 1 = 1.) See also [HMP20] for a general discussion regarding "profinitely rigid" elements in finitely generated groups and the relation to measures induced by such elements on finite groups.…”

Section: Measures Induced By Elements Of Finitely Generated Groupsmentioning

confidence: 99%

Local Statistics of Random Permutations from Free Products

Puder¹,

Zimhoni²

2022

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Let Γ = G 1 * . . . * G k be a free product of groups where each of G 1 , . . . , G k is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element γ ∈ Γ, a γ-random permutation in the symmetric group S N is the image of γ through a uniformly random homomorphism Γ → S N . In this paper we study local statistics of γ-random permutations and their asymptotics as N grows. We first consider E [fix γ (N )], the expected number of fixed points in a γ-random permutation in S N . We show that unless γ has finite order, the limit of E [fix γ (N )] as N → ∞ is an integer, and is equal to the number of subgroups H ≤ Γ containing γ such that H ∼ = Z or H ∼ = C 2 * C 2 . Equivalently, this is the number of subgroups H ≤ Γ containing γ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for E [fix γ (N )] and determine the limit distribution of the number of fixed points as N → ∞. These results are then generalized to all statistics of cycles of fixed lengths.

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“…However, since the Lie groups they consider are sometimes infinite, this does not imply that the set of surface words is separable. Hanany, Meiri and Puder proved that the automorphism orbit of the commutator [a 1 , a 2 ] is separable [7], again by studying the push-forward measure associated to a random map to a symmetric group.…”

Section: Question 3 Let W ∈ F Is the Automorphism Orbit Aut(f )W Sep...mentioning

confidence: 99%