On the distribution of values of certain word maps
HTML articles powered by AMS MathViewer
- by Michael Larsen and Aner Shalev PDF
- Trans. Amer. Math. Soc. 368 (2016), 1647-1661 Request permission
Abstract:
We prove that, for any positive integers $m, n$, the word map $(x,y) \mapsto x^my^n$ is almost measure preserving on finite simple groups. This extends the case $m=n=2$ obtained in 2009. Along the way we obtain results of independent interest on fibers of word maps and on character values.References
- Tatiana Bandman and Shelly Garion, Surjectivity and equidistribution of the word $x^a y^b$ on $\textrm {PSL}(2,q)$ and $\textrm {SL}(2,q)$, Internat. J. Algebra Comput. 22 (2012), no.Β 2, 1250017, 33. MR 2903751, DOI 10.1142/S0218196712500178
- A. K. Das and R. K. Nath, On solutions of a class of equations in a finite group, Comm. Algebra 37 (2009), no.Β 11, 3904β3911. MR 2573226, DOI 10.1080/00927870902828777
- Shelly Garion, Michael Larsen, and Alexander Lubotzky, Beauville surfaces and finite simple groups, J. Reine Angew. Math. 666 (2012), 225β243. MR 2920887, DOI 10.1515/CRELLE.2011.117
- Shelly Garion and Aner Shalev, Commutator maps, measure preservation, and $T$-systems, Trans. Amer. Math. Soc. 361 (2009), no.Β 9, 4631β4651. MR 2506422, DOI 10.1090/S0002-9947-09-04575-9
- Michael Larsen and Aner Shalev, Characters of symmetric groups: sharp bounds and applications, Invent. Math. 174 (2008), no.Β 3, 645β687. MR 2453603, DOI 10.1007/s00222-008-0145-7
- Michael Larsen and Aner Shalev, Fibers of word maps and some applications, J. Algebra 354 (2012), 36β48. MR 2879221, DOI 10.1016/j.jalgebra.2011.10.040
- Michael Larsen, Aner Shalev, and Pham Huu Tiep, The Waring problem for finite simple groups, Ann. of Math. (2) 174 (2011), no.Β 3, 1885β1950. MR 2846493, DOI 10.4007/annals.2011.174.3.10
- Martin W. Liebeck and Aner Shalev, Diameters of finite simple groups: sharp bounds and applications, Ann. of Math. (2) 154 (2001), no.Β 2, 383β406. MR 1865975, DOI 10.2307/3062101
- Martin W. Liebeck and Aner Shalev, Fuchsian groups, finite simple groups and representation varieties, Invent. Math. 159 (2005), no.Β 2, 317β367. MR 2116277, DOI 10.1007/s00222-004-0390-3
- Martin W. Liebeck and Aner Shalev, Character degrees and random walks in finite groups of Lie type, Proc. London Math. Soc. (3) 90 (2005), no.Β 1, 61β86. MR 2107038, DOI 10.1112/S0024611504014935
- Martin W. Liebeck, E. A. OβBrien, Aner Shalev, and Pham Huu Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), no.Β 4, 939β1008. MR 2654085, DOI 10.4171/JEMS/220
- A. M. Macbeath, Generators of the linear fractional groups, Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967) Amer. Math. Soc., Providence, R.I., 1969, pp.Β 14β32. MR 0262379
- Rajat Kanti Nath, A new class of almost measure preserving maps on finite simple groups, J. Algebra Appl. 13 (2014), no.Β 4, 1350142, 5. MR 3153877, DOI 10.1142/S0219498813501429
- Doron Puder and Ori Parzanchevski, Measure preserving words are primitive, J. Amer. Math. Soc. 28 (2015), no.Β 1, 63β97. MR 3264763, DOI 10.1090/S0894-0347-2014-00796-7
- Ori Parzanchevski and Gili Schul, On the Fourier expansion of word maps, Bull. Lond. Math. Soc. 46 (2014), no.Β 1, 91β102. MR 3161765, DOI 10.1112/blms/bdt068
- Gili Schul and Aner Shalev, Words and mixing times in finite simple groups, Groups Geom. Dyn. 5 (2011), no.Β 2, 509β527. MR 2782183, DOI 10.4171/GGD/137
- Dan Segal, Words: notes on verbal width in groups, London Mathematical Society Lecture Note Series, vol. 361, Cambridge University Press, Cambridge, 2009. MR 2547644, DOI 10.1017/CBO9781139107082
- Aner Shalev, Mixing and generation in simple groups, J. Algebra 319 (2008), no.Β 7, 3075β3086. MR 2397424, DOI 10.1016/j.jalgebra.2007.07.031
- Aner Shalev, Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. of Math. (2) 170 (2009), no.Β 3, 1383β1416. MR 2600876, DOI 10.4007/annals.2009.170.1383
- Aner Shalev, Applications of some zeta functions in group theory, Zeta functions in algebra and geometry, Contemp. Math., vol. 566, Amer. Math. Soc., Providence, RI, 2012, pp.Β 331β344. MR 2858929, DOI 10.1090/conm/566/11227
- Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
Additional Information
- Michael Larsen
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 293592
- Email: mjlarsen@indiana.edu
- Aner Shalev
- Affiliation: Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel
- MR Author ID: 228986
- ORCID: 0000-0001-9428-2958
- Email: shalev@math.huji.ac.il
- Received by editor(s): August 6, 2013
- Received by editor(s) in revised form: December 26, 2013
- Published electronically: May 13, 2015
- Additional Notes: The first author was partially supported by NSF Grant DMS-1101424. The second author was partially supported by ERC Advanced Grant 247034 and the Vinik Chair in Mathematics. Both authors were partially supported by Binational Science Foundation United States-Israel Grant 2008194.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 1647-1661
- MSC (2010): Primary 20P05, 20D06
- DOI: https://doi.org/10.1090/tran/6389
- MathSciNet review: 3449221