Measure preserving words are primitive
By Doron Puder, Ori Parzanchevski. 2012
87 citations recorded by the source corpus.
A word w 2 Fk, the free group on k generators, is called primitive if it belongs to some basis of Fk. Associated with w and a finite group G is the word map w : G×...×G ! G defined on the direct product of k copies of G. We call w measure preserving if given uniform measure on G×...×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 Mobius inversions. Our methods yield the stronger result that a subgroup of Fk 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 Fk form a closed set in this topology.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5f939c16e1b892e48d511daf8a8888b07cedebff
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In Computational Theory and Mathematics
In Geometry and Topology
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Citations in the corpus
- Expansion of random graphs: new proofs, new results (2012, methodology): “) The main theorem in [53] estimates the expected number of fixed points of the random permutation w (σ1, .”