Measure preserving words are primitive

Doron Puder*, Ori Parzanchevski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Scopus citations

Abstract

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, a word w in the free group on K generators induces a word map from Gk to G. We say that w is measure preserving with respect to G if given uniform distribution on GK, the image of this word map distributes uniformly on 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 Fk 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 Fk form a closed set in this topology.

Original languageAmerican English
Pages (from-to)63-97
Number of pages35
JournalJournal of the American Mathematical Society
Volume28
Issue number1
DOIs
StatePublished - 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2014, American Mathematical Society.

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