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Surface Words are Determined by Word Measures on Groups

Magee, Michael; Puder, Doron

Surface Words are Determined by Word Measures on Groups Thumbnail


Doron Puder


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 1ψ(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.


Magee, M., & Puder, D. (2021). Surface Words are Determined by Word Measures on Groups. Israel Journal of Mathematics, 241, 749-774.

Journal Article Type Article
Acceptance Date Feb 7, 2020
Online Publication Date Mar 6, 2021
Publication Date 2021-03
Deposit Date Mar 27, 2020
Publicly Available Date Mar 6, 2022
Journal Israel Journal of Mathematics
Print ISSN 0021-2172
Electronic ISSN 1565-8511
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 241
Pages 749-774


Accepted Journal Article (468 Kb)

Copyright Statement
This is a post-peer-review, pre-copyedit version of an article published in Israel journal of mathematics. The final authenticated version is available online at:

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