Primitive Words, Free Factors and Measure Preservation

Details Primitive Words, Free Factors and Measure Preservation Primitive Words, Free Factors and Measure Preservation

2011-04-20

arxiv.org/abs/1104.3991v3

Mathematics

Group Theory

This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Pre

Scientific paper

Let F_k be the free group on k generators, and let H \le J \le F_k be subgroups of finite rank. Using Stallings core graphs, we give a simple graph-theoretic criterion to determine whether H is a free factor of J. In particular, this criterion determines whether a given element w in F_k is primitive, i.e. whether it belongs to some basis of F_k. This translates immediately to an algorithm to detect primitive words and free factor subgroups of F_k. We also deal with a question related to measure preservation. It is conjectured that a word w in F_k is primitive iff it is measure preserving in the following sense. Let G be an arbitrary finite group. Consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G). We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). That primitivity implies measure preservation is easy to show, but the converse is still unknown. We study this problem and in particular prove the conjecture for k=2. Both primitivity and measure preservation can be naturally defined as well for finite subsets of F_k. Our proof in fact shows that for sets of cardinality \ge k-1 in F_k the two notions coincide. The above conjecture implies a positive answer to a question regarding the profinite topology of free groups: Do the primitive elements of F_k form a closed set in this topology? Our results translate naturally to this context.

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