Word maps and Waring type problems

Larsen, Michael; Shalev, Aner

doi:10.1090/S0894-0347-08-00615-2

Word maps and Waring type problems
HTML articles powered by AMS MathViewer

by Michael Larsen and Aner Shalev;

J. Amer. Math. Soc. 22 (2009), 437-466

DOI: https://doi.org/10.1090/S0894-0347-08-00615-2

Published electronically: September 12, 2008

PDF | Request permission

Abstract:

Waring’s classical problem deals with expressing every natural number as a sum of $g(k)$ $k$th powers. Recently there has been considerable interest in similar questions for nonabelian groups and simple groups in particular. Here the $k$th power word is replaced by an arbitrary group word $w \ne 1$, and the goal is to express group elements as short products of values of $w$.

We give a best possible and somewhat surprising solution for this Waring type problem for various finite simple groups, showing that a product of length two suffices to express all elements. We also show that the set of values of $w$ is very large, improving various results obtained so far.

Along the way we also obtain new results of independent interest on character values and class squares in symmetric groups.

Our methods involve algebraic geometry, representation theory, probabilistic arguments, as well as results from analytic number theory, including three primes theorems (approximating Goldbach’s Conjecture).

References

Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 354654
Z. Arad and M. Herzog (eds.), Products of conjugacy classes in groups, Lecture Notes in Mathematics, vol. 1112, Springer-Verlag, Berlin, 1985. MR 783067, DOI 10.1007/BFb0072284
Raymond Ayoub, On Rademacher’s extension of the Goldbach-Vinogradoff theorem, Trans. Amer. Math. Soc. 74 (1953), 482–491. MR 53960, DOI 10.1090/S0002-9947-1953-0053960-0
R. C. Baker, G. Harman, and J. Pintz, The exceptional set for Goldbach’s problem in short intervals, Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995) London Math. Soc. Lecture Note Ser., vol. 237, Cambridge Univ. Press, Cambridge, 1997, pp. 1–54. MR 1635718, DOI 10.1017/CBO9780511526091.004
Edward Bertram, Even permutations as a product of two conjugate cycles, J. Combinatorial Theory Ser. A 12 (1972), 368–380. MR 297853, DOI 10.1016/0097-3165(72)90102-1
A. Borel, On free subgroups of semisimple groups, Enseign. Math. (2) 29 (1983), no. 1-2, 151–164. MR 702738
J. L. Brenner, Covering theorems for finite nonabelian simple groups. IX. How the square of a class with two nontrivial orbits in $S_{n}$ covers $A_{n}$, Ars Combin. 4 (1977), 151–176. MR 576549
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520, DOI 10.1007/BF02684780
Erich W. Ellers, Nikolai Gordeev, and Marcel Herzog, Covering numbers for Chevalley groups, Israel J. Math. 111 (1999), 339–372. MR 1710745, DOI 10.1007/BF02810691
P. Erdős and P. Turán, On some problems of a statistical group-theory. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 175–186 (1965). MR 184994, DOI 10.1007/BF00536750
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
Dale H. Husemoller, Ramified coverings of Riemann surfaces, Duke Math. J. 29 (1962), 167–174. MR 136726
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828, DOI 10.1007/BFb0067708
Gareth A. Jones, Varieties and simple groups, J. Austral. Math. Soc. 17 (1974), 163–173. MR 344342, DOI 10.1017/S1446788700016748
Serge Lang, Sur les séries $L$ d’une variété algébrique, Bull. Soc. Math. France 84 (1956), 385–407 (French). MR 88777, DOI 10.24033/bsmf.1477
Michael Larsen, Word maps have large image, Israel J. Math. 139 (2004), 149–156. MR 2041227, DOI 10.1007/BF02787545
R. Lawther and Martin W. Liebeck, On the diameter of a Cayley graph of a simple group of Lie type based on a conjugacy class, J. Combin. Theory Ser. A 83 (1998), no. 1, 118–137. MR 1629452, DOI 10.1006/jcta.1998.2869
Martin W. Liebeck and Aner Shalev, Diameters of finite simple groups: sharp bounds and applications, Ann. of Math. (2) 154 (2001), no. 2, 383–406. MR 1865975, DOI 10.2307/3062101
Martin W. Liebeck and Aner Shalev, Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks, J. Algebra 276 (2004), no. 2, 552–601. MR 2058457, DOI 10.1016/S0021-8693(03)00515-5
Martin W. Liebeck and Aner Shalev, Fuchsian groups, finite simple groups and representation varieties, Invent. Math. 159 (2005), no. 2, 317–367. MR 2116277, DOI 10.1007/s00222-004-0390-3
C. Martinez and E. Zelmanov, Products of powers in finite simple groups. part B, Israel J. Math. 96 (1996), no. part B, 469–479. MR 1433702, DOI 10.1007/BF02937318
Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 266911
Thomas W. Müller and Jan-Christoph Schlage-Puchta, Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks, Adv. Math. 213 (2007), no. 2, 919–982. MR 2332616, DOI 10.1016/j.aim.2007.01.016
Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. The classical bases. MR 1395371, DOI 10.1007/978-1-4757-3845-2
Nikolay Nikolov and Dan Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. of Math. (2) 165 (2007), no. 1, 171–238. MR 2276769, DOI 10.4007/annals.2007.165.171
Nikolay Nikolov and Dan Segal, On finitely generated profinite groups. II. Products in quasisimple groups, Ann. of Math. (2) 165 (2007), no. 1, 239–273. MR 2276770, DOI 10.4007/annals.2007.165.239
Richard Pink, Compact subgroups of linear algebraic groups, J. Algebra 206 (1998), no. 2, 438–504. MR 1637068, DOI 10.1006/jabr.1998.7439
Jan Saxl and John S. Wilson, A note on powers in simple groups, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, 91–94. MR 1443588, DOI 10.1017/S030500419600165X

Annals of Math.

J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, RI, 1966, pp. 33–62. MR 224710
John Wilson, First-order group theory, Infinite groups 1994 (Ravello), de Gruyter, Berlin, 1996, pp. 301–314. MR 1477188