On the Fourier expansion of word maps

Parzanchevski, Ori; Schul, Gili

doi:10.1112/blms/bdt068

Cited by 19 publications

(19 citation statements)

References 4 publications

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“…They belong to different Aut F 2 -orbits, as w 2 ∈ F 2 but w 1 / ∈ F 2 , but induce the same distribution on S n for every n: their images under a random homomorphism are a product of a random permutation (σ) and a random element in its conjugacy class (τ στ −1 for w 1 , and τ σ −1 τ −1 for w 2 ). However, while S n do not distinguish between these two words, other groups do (in fact these words induce the same distribution on G precisely when every element in G is conjugate to its inverse, see [PS14] for a discussion of this).…”

Section: The Analysis Of C X Mnmentioning

confidence: 99%

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Measure preserving words are primitive

Puder

Parzanchevski

2014

J. Amer. Math. Soc.

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We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group G G , a word w w in the free group on k k generators induces a word map from G k G^{k} to G G . We say that w w is measure preserving with respect to G G if given uniform distribution on G k G^{k} , the image of this word map distributes uniformly on G G . It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. Here we prove this conjecture. The main ingredients of the proof include random coverings of Stallings graphs, algebraic extensions of free groups, and Möbius inversions. Our methods yield the stronger result that a subgroup of F k \mathbf {F}_{k} is measure preserving if and only if it is a free factor. As an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of F k \mathbf {F}_{k} form a closed set in this topology.

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Section: The Analysis Of C X Mnmentioning

confidence: 99%

“…The last years have seen a great interest in word maps in groups, and the distributions they induce. We refer the reader, for instance, to [Sha09, LS09, AV11,PS14], and to the recent book [Seg09] and survey [Sha13]. Several authors have also studied words which are asymptotically measure preserving on finite simple groups, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Measure preserving words are primitive

Puder

Parzanchevski

2014

J. Amer. Math. Soc.

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“…The reason is of course that N ww ′ = N w * N w ′ is the convolution of the two functions N w and N w ′ . More information in this direction is given in [10].…”

Section: Words In P-groups Of Nilpotency Classmentioning

confidence: 99%

“…But for any k, π maps 2 w k onto G w k , thus G w k = G ′ for k ≥ d 0 . Now it is easy to show that if (10) So in order to prove (10) for w k we can always assume that 1 ≤ k ≤ d 0 .…”

Section: P-groups With Central Frattini Subgroupmentioning

confidence: 99%

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Words and characters in finite p-groups

Iñiguez¹,

Sangroniz²

2017

Journal of Algebra

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Given a group word w in k variables, a finite group G and g ∈ G, we consider the number N w,G (g) of k-tuples g 1 , . . . , g k of elements of G such that w(g 1 , . . . , g k ) = g. In this work we study the functions N w,G for the class of nilpotent groups of nilpotency class 2. We show that, for the groups in this class, N w,G (1) ≥ |G| k−1 , an inequality that can be improved to N w,G (1) ≥ |G| k /|G w | (G w is the set of values taken by w on G) if G has odd order. This last result is explained by the fact that the functions N w,G are characters of G in this case. For groups of even order, all that can be said is that N w,G is a generalized character, something that is false in general for groups of nilpotency class greater than 2. We characterize group theoretically when N x n ,G is a character if G is a 2-group of nilpotency class 2. Finally we also address the (much harder) problem of studying if N w,G (g) ≥ |G| k−1 for g ∈ G w , proving that this is the case for the free p-groups of nilpotency class 2 and exponent p.

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Matrix group integrals, surfaces, and mapping class groups II: $$\textrm{O}\left( n\right) $$ and $$\textrm{Sp}\left( n\right) $$

Magee

Puder

2022

Math. Ann.

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On the Fourier expansion of word maps

Cited by 19 publications

References 4 publications

Measure preserving words are primitive

Measure preserving words are primitive

Words and characters in finite p-groups

Matrix group integrals, surfaces, and mapping class groups II: $$\textrm{O}\left( n\right) $$ and $$\textrm{Sp}\left( n\right) $$

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