Measure preserving words are primitive

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 groupGG, a wordwwin the free group onkkgenerators induces aword mapfromGkG^{k}toGG. We say thatwwis measure preserving with respect toGGif given uniform distribution onGkG^{k}, the image of this word map distributes uniformly onGG. 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 ofFk\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 ofFk\mathbf {F}_{k}form a closed set in this topology.