Measure Preserving Words are Primitive

Details Measure Preserving Words are Primitive Measure Preserving Words are Primitive

2012-02-15

arxiv.org/abs/1202.3269v1

Mathematics

Group Theory

39 pages, 6 figures

Scientific paper

A word w \in F_k, the free group on k generators, is called primitive if it belongs to some basis of F_k. Associated with w and a finite group G is the word map w: G x G x ... x G \to G defined on the direct product of k copies of G. We call w measure preserving if given uniform measure on G x G x ... x G, the image of this word map induces uniform measure on G (for every finite group G). It is easy to see that every primitive word is measure preserving, 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 M\"obius inversions. Our methods yield the stronger result that a subgroup of F_k is measure preserving iff 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 form a closed set in this topology.

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