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Measure preserving words are primitive. (English) Zbl 1402.20042

Summary: 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 \(G^k\) to \( G\). We say that \(w\) is measure preserving with respect to \(G\) if given uniform distribution on \(G^k\), 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 \(\mathbf {F}_k\) 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 \(\mathbf {F}_k\) form a closed set in this topology.

MSC:

20E05 Free nonabelian groups
20E18 Limits, profinite groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E22 Extensions, wreath products, and other compositions of groups

Software:

GAP; FGA
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References:

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