Measure preserving words are primitive

Puder, Doron¹ ; Parzanchevski, Ori²

¹ Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
² School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Journal of the American Mathematical Society, Tome 28 (2015) no. 1, pp. 63-97

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-2014-00796-7

Bibliographie

[1] Abã©Rt, Miklã³S On the probability of satisfying a word in a group J. Group Theory 2006 685 694

[2] Amit, Alon, Linial, Nathan Random graph coverings. I. General theory and graph connectivity Combinatorica 2002 1 18

[3] Alon, N. Eigenvalues and expanders Combinatorica 1986 83 96

[4] Amit, Alon, Vishne, Uzi Characters and solutions to equations in finite groups J. Algebra Appl. 2011 675 686

[5] Bandman, Tatiana, Kunyavskiä­, Boris Criteria for equidistribution of solutions of word equations on 𝑆𝐿(2) J. Algebra 2013 282 302

[6] Broder, Andrei, Eli Shamir On the second eigenvalue of random regular graphs Foundations of Computer Science 1987

[7] Friedman, Joel Relative expanders or weakly relatively Ramanujan graphs Duke Math. J. 2003 19 35

[8] Joel Friedman A proof of Alon’s second eigenvalue conjecture and related problems Memoirs of the AMS September 2008

[9] The Gap Group GAP – Groups, Algorithms, and Programming, Version 4.6.5 2013

[10] Garion, Shelly, Shalev, Aner Commutator maps, measure preservation, and 𝑇-systems Trans. Amer. Math. Soc. 2009 4631 4651

[11] Kapovich, Ilya, Myasnikov, Alexei Stallings foldings and subgroups of free groups J. Algebra 2002 608 668

[12] Linial, Nati, Puder, Doron Word maps and spectra of random graph lifts Random Structures Algorithms 2010 100 135

[13] Larsen, Michael, Shalev, Aner Characters of symmetric groups: sharp bounds and applications Invent. Math. 2008 645 687

[14] Larsen, Michael, Shalev, Aner Word maps and Waring type problems J. Amer. Math. Soc. 2009 437 466

[15] Miasnikov, Alexei, Ventura, Enric, Weil, Pascal Algebraic extensions in free groups 2007 225 253

[16] Nica, Alexandru On the number of cycles of given length of a free word in several random permutations Random Structures Algorithms 1994 703 730

[17] Ori Parzanchevski, Doron Puder Stallings graphs, algebraic extensions and primitive elements in 𝐹₂ Mathematical Proceedings of the Cambridge Philosophical Society 2014

[18] Ori Parzanchevski, Gili Schul On the Fourier expansion of word maps Bull. London Math. Soc. 2014 91 102

[19] Doron Puder Expansion of random graphs: New proofs, new results 2012

[20] Doron Puder Primitive words, free factors and measure preservation Israel Journal of Mathematics 2013

[21] Doron Puder And Conan Wu Growth of primitives elements in free groups Journal of London Mathematical Society 2014

[22] Segal, Dan Words: notes on verbal width in groups 2009

[23] Shalev, Aner Word maps, conjugacy classes, and a noncommutative Waring-type theorem Ann. of Math. (2) 2009 1383 1416

[24] Aner Shalev Some results and problems in the theory of word maps 2013

[25] Christian Sievers Free Group Algorithms – a GAP package, Version 1.2.0 2012

[26] Stallings, John R. Topology of finite graphs Invent. Math. 1983 551 565

[27] Stanley, Richard P. Enumerative combinatorics. Vol. 1 1997

[28] Takahasi, Mutuo Note on chain conditions in free groups Osaka Math. J. 1951 221 225

[29] Turner, Edward C. Test words for automorphisms of free groups Bull. London Math. Soc. 1996 255 263

[30] Van Lint, J. H., Wilson, R. M. A course in combinatorics 2001

[31] Whitehead, J. H. C. On Certain Sets of Elements in a Free Group Proc. London Math. Soc. (2) 1936 48 56

[32] John H. C. Whitehead On equivalent sets of elements in a free group Ann. of Math. 1936 768 800

[33] Wilson, John S. Profinite groups 1998