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Geometry of word equations in simple algebraic groups over special fields. (English. Russian original) Zbl 1442.20028

Russ. Math. Surv. 73, No. 5, 753-796 (2018); translation from Usp. Mat. Nauk 73, No. 5, 3-52 (2018).
Authors’ abstract: This paper contains a survey of recent developments in the investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras, along with some new results on the images of word maps on algebraic groups defined over special fields: complex, real, \(p\)-adic (or close to such), or finite.

MSC:

20F70 Algebraic geometry over groups; equations over groups
20G15 Linear algebraic groups over arbitrary fields
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)

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References:

[1] M. Aka, E. Breuillard, L. Rosenzweig, and N. de Saxcé 2015 Diophantine properties of nilpotent Lie groups Compos. Math.151 6 1157-1188 · Zbl 1332.22011 · doi:10.1112/S0010437X14007854
[2] D. Akhiezer 2015 On the commutator map for real semisimple Lie algebras Mosc. Math. J.15 4 609-613 · Zbl 1352.17008
[3] A. Amit and U. Vishne 2011 Characters and solutions to equations in finite groups J. Algebra Appl.10 4 675-686 · Zbl 1246.20030 · doi:10.1142/S0219498811004690
[4] B. E. Anzis, Z. M. Emrich, and K. G. Valiveti 2015 On the images of Lie polynomials evaluated on Lie algebras Linear Algebra Appl.469 51-75 · Zbl 1366.17004 · doi:10.1016/j.laa.2014.11.015
[5] N. Avni, T. Gelander, M. Kassabov, and A. Shalev 2013 Word values in \(p\)-adic and adelic groups Bull. Lond. Math. Soc.45 6 1323-1330 · Zbl 1288.20062 · doi:10.1112/blms/bdt063
[6] T. Bandman, S. Garion, and F. Grunewald 2012 On the surjectivity of Engel words on \(\operatorname{PSL}(2,q)\) Groups Geom. Dyn.6 3 409-439 · Zbl 1261.14010 · doi:10.4171/GGD/162
[7] T. Bandman, S. Garion, and B. Kunyavskiĭ 2014 Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics Cent. Eur. J. Math.12 2 175-211 · Zbl 1294.20056 · doi:10.2478/s11533-013-0335-4
[8] T. Bandman, N. Gordeev, B. Kunyavskiĭ, and E. Plotkin 2012 Equations in simple Lie algebras J. Algebra355 67-79 · Zbl 1297.17003 · doi:10.1016/j.jalgebra.2012.01.012
[9] T. Bandman and B. Kunyavskiĭ 2013 Criteria for equidistribution of solutions of word equations on \(\operatorname{SL}(2) J. Algebra 382 282-302\) · Zbl 1292.20049 · doi:10.1016/j.jalgebra.2013.02.031
[10] T. Bandman and Yu. G. Zarhin 2016 Surjectivity of certain word maps on \(\operatorname{PSL}(2,\mathbb C)\) and \(\operatorname{SL}(2,\mathbb C)\) Eur. J. Math.2 3 614-643 · Zbl 1392.20038 · doi:10.1007/s40879-016-0101-9
[11] J. Barge and E. Ghys 1992 Cocycles d’Euler et de Maslov Math. Ann.294 2 235-265 · Zbl 0894.55006 · doi:10.1007/BF01934324
[12] G. M. Bergman and N. Nahlus 2011 Homomorphisms on infinite direct product algebras, especially Lie algebras J. Algebra333 67-104 · Zbl 1276.17002 · doi:10.1016/j.jalgebra.2011.02.035
[13] Y. Billig and V. Futorny 2018 Lie algebras of vector fields on smooth affine varieties Comm. Algebra46 8 3413-3429 · Zbl 1434.17029 · doi:10.1080/00927872.2017.1412456
[14] J. Blanc 2010 Groupes de Cremona, connexité et simplicité Ann. Sci. Éc. Norm. Supér. (4)43 2 357-364 · Zbl 1193.14017 · doi:10.24033/asens.2123
[15] J. Blanc and J.-P. Furter 2013 Topologies and structures of the Cremona groups Ann. of Math. (2)178 3 1173-1198 · Zbl 1298.14020 · doi:10.4007/annals.2013.178.3.8
[16] J. Blanc and S. Zimmermann 2018 Topological simplicity of the Cremona groups Amer. J. Math.140 5 · Zbl 1453.14041
[17] J. Blanc and S. Zimmermann 1511.08907
[18] J.-M. Bois 2009 Generators of simple Lie algebras in arbitrary characteristics Math. Z.262 4 715-741 · Zbl 1170.17004 · doi:10.1007/s00209-008-0397-3
[19] A. Borel 1983 On free subgroups of semi-simple groups Enseign. Math. (2)29 1-2 151-164 · Zbl 0533.22009
[20] A. Borel 2001 Œuvres: collected papers IV Springer-Verlag, Berlin 41-54
[21] A. Borel 1991 Linear algebraic groups Grad. Texts in Math. 126 Springer-Verlag, New York 2nd ed., xii+288 pp. · doi:10.1007/978-1-4612-0941-6
[22] A. Borel 1997 Class functions, conjugacy classes and commutators in semisimple Lie groups Algebraic groups and Lie groups Austral. Math. Soc. Lect. Ser. 9 Cambridge Univ. Press, Cambridge 1-19 · Zbl 0872.22011
[23] A. Bors 2017 Fibers of word maps and the multiplicities of non-abelian composition factors Internat. J. Algebra Comput.27 8 1121-1148 · Zbl 1483.20030 · doi:10.1142/S0218196717500539
[24] Kh. Bou-Rabee and M. Larsen 2017 Linear groups with Borel’s property J. Eur. Math. Soc. (JEMS)19 5 1293-1330 · Zbl 1475.20048 · doi:10.4171/JEMS/693
[25] N. Bourbaki 1981 Éléments de mathématique. Groupes et algèbres de Lie, Ch. 4-6 Masson, Paris 2 ème éd., 290 pp.
[26] E. Breuillard, B. Green, R. Guralnick, and T. Tao 2012 Strongly dense free subgroups of semisimple algebraic groups Israel J. Math.192 1 347-379 · Zbl 1266.20060 · doi:10.1007/s11856-012-0030-3
[27] G. Brown 1963 On commutators in a simple Lie algebra Proc. Amer. Math. Soc.14 763-767 · Zbl 0135.07304 · doi:10.1090/S0002-9939-1963-0153715-4
[28] D. Burago, S. Ivanov, and L. Polterovich 2008 Conjugation-invariant norms on groups of geometric origin Groups of diffeomorphisms Adv. Stud. Pure Math. 52 Math. Soc. Japan, Tokyo 221-250 · Zbl 1222.20031
[29] D. Buzinski and R. Winstanley 2013 On multilinear polynomials in four variables evaluated on matrices Linear Algebra Appl.439 9 2712-2719 · Zbl 1283.15053 · doi:10.1016/j.laa.2013.08.005
[30] D. Calegari and D. Zhuang 2011 Stable \(W\)-length Topology and geometry in dimension three Contemp. Math. 560 Amer. Math. Soc., Providence, RI 145-169 · Zbl 1333.20045 · doi:10.1090/conm/560/11097
[31] S. Cantat and S. Lamy 2013 Normal subgroups in the Cremona group Acta Math.210 31-94 with an appendix by Y. de Cornulier · Zbl 1278.14017 · doi:10.1007/s11511-013-0090-1
[32] P.-E. Caprace and K. Fujiwara 2010 Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups Geom. Funct. Anal.19 5 1296-1319 · Zbl 1206.20046 · doi:10.1007/s00039-009-0042-2
[33] P. Chatterjee 2002 On the surjectivity of the power maps of algebraic groups in characteristic zero Math. Res. Lett.9 5-6 741-756 · Zbl 1025.20034 · doi:10.4310/MRL.2002.v9.n6.a4
[34] P. Chatterjee 2003 On the surjectivity of the power maps of semisimple algebraic groups Math. Res. Lett.10 5-6 625-633 · Zbl 1045.20043 · doi:10.4310/MRL.2003.v10.n5.a6
[35] P. Chatterjee 2009 On the power maps, orders and exponentiality of \(p\)-adic algebraic groups J. Reine Angew. Math.2009 629 201-220 · Zbl 1190.20040 · doi:10.1515/CRELLE.2009.031
[36] P. Chatterjee 2011 Surjectivity of power maps of real algebraic groups Adv. Math.226 6 4639-4666 · Zbl 1217.22004 · doi:10.1016/j.aim.2010.11.005
[37] W. Cocke and M.-C. Ho 2018 On the symmetry of images of word maps in groups Comm. Algebra46 2 756-763 · Zbl 1387.20025 · doi:10.1080/00927872.2017.1327065
[38] B. Conrad 2014 Reductive group schemes Autour des schémas en groupes Panor. Synthèses 42/43, I Soc. Math. France, Paris 93-444 · Zbl 1349.14151
[39] A. D’Andrea and A. Maffei 2016 Commutators of small elements in compact semisimple groups and Lie algebras J. Lie Theory26 3 683-690 · Zbl 1348.22009
[40] D. Ž. Đoković and K. H. Hofmann 1997 The surjectivity question for the exponential function of real Lie groups: a status report J. Lie Theory7 2 171-199 · Zbl 0888.22003
[41] D. Ž. Đoković and T.-Y. Tam 2003 Some questions about semisimple Lie groups originating in matrix theory Canad. Math. Bull.46 3 332-343 · Zbl 1047.22013 · doi:10.4153/CMB-2003-035-1
[42] D. Ž. Đoković and N. Q. Thăńg 1995 On the exponential map of almost simple real algebraic groups J. Lie Theory5 2 275-291 · Zbl 0851.20046
[43] S. K. Donaldson 2011 Lectures on Lie groups and geometry, Notes for course given in 2007 and 2011 95 pp. http://wwwf.imperial.ac.uk/ skdona/LIEGROUPS2011.PDF
[44] N. M. Dunfield and W. P. Thurston 2006 Finite covers of random 3-manifolds Invent. Math.166 3 457-521 · Zbl 1111.57013 · doi:10.1007/s00222-006-0001-6
[45] K. J. Dykema and I. Klep 2016 Instances of the Kaplansky-Lvov multilinear conjecture for polynomials of degree three Linear Algebra Appl.508 272-288 · Zbl 1361.16012 · doi:10.1016/j.laa.2016.08.005
[46] K. Dykema and A. Skripka 2012 On single commutators in \(II_1\)-factors Proc. Amer. Math. Soc.140 3 931-940 · Zbl 1273.47063 · doi:10.1090/S0002-9939-2011-10953-5
[47] A. Elkasapy and A. Thom 2014 About Gotô’s method showing surjectivity of word maps Indiana Univ. Math. J.63 5 1553-1565 · Zbl 1320.20033 · doi:10.1512/iumj.2014.63.5391
[48] A. Elkasapy and A. Thom 2015 On the length of the shortest non-trivial element in the derived and the lower central series J. Group Theory18 5 793-804 · Zbl 1346.20045 · doi:10.1515/jgth-2015-0007
[49] E. W. Ellers and N. Gordeev 1998 On the conjectures of J. Thompson and O. Ore Trans. Amer. Math. Soc.350 9 3657-3671 · Zbl 0910.20007 · doi:10.1090/S0002-9947-98-01953-9
[50] E. W. Ellers and N. Gordeev 2004 Intersection of conjugacy classes with Bruhat cells in Chevalley groups Pacific J. Math.214 2 245-261 · Zbl 1062.20050 · doi:10.2140/pjm.2004.214.245
[51] E. Fink and A. Thom 2015 Palindromic words in simple groups Internat. J. Algebra Comput.25 3 439-444 · Zbl 1321.20028 · doi:10.1142/S0218196715500046
[52] A. Galt, A. Kulshrestha, A. Singh, and E. Vdovin 2018 On Shalev’s conjecture for type \(A_n\) and \({}^2A_n\) 1805.04638 16 pp.
[53] J.-M. Gambaudo and É. Ghys 2004 Commutators and diffeomorphisms of surfaces Ergodic Theory Dynam. Systems24 5 1591-1617 · Zbl 1088.37018 · doi:10.1017/S0143385703000737
[54] S. Garion and A. Shalev 2009 Commutator maps, measure preservation, and \(T\)-systems Trans. Amer. Math. Soc.361 9 4631-4651 · Zbl 1182.20015 · doi:10.1090/S0002-9947-09-04575-9
[55] С. И. Гельфанд 2004 О числе решений квадратного уравнения Глобус 1, Общематический семинарНМУ, 2000 МЦНМО, М. 124-133
[56] S. I. Gelfand 2004 The number of solutions of a quadratic equation Globus 1, General mathematical seminarIndependent University of Moscow 2000 Moscow Centre for Continuous Mathematical Education, Moscow 124-133
[57] M. Gerstenhaber and O. S. Rothaus 1962 The solution of sets of equations in groups Proc. Nat. Acad. Sci. U.S.A.48 9 1531-1533 · Zbl 0112.02504 · doi:10.1073/pnas.48.9.1531
[58] N. L. Gordeev 1995 Products of conjugacy classes in algebraic groups. I J. Algebra173 3 715-744 · Zbl 0832.14037 · doi:10.1006/jabr.1995.1111
[59] N. L. Gordeev 1997 Freedom in conjugacy classes of simple algebraic groups and identities with constants Алгебра и анализ9 4 63-78 · Zbl 0897.20034
[60] N. L. Gordeev 1998 St. Petersburg Math. J.9 4 709-723
[61] N. L. Gordeev 2005 Products of conjugacy classes in perfect linear groups. Extended covering number Вопросы теории представлений алгебр и групп. 12 Зап. науч. сем. ПОМИ 321 ПОМИ, СПб. 67-89 · Zbl 1088.20024
[62] N. L. Gordeev 2006 J. Math. Sci. (N. Y.)136 3 3867-3879 · Zbl 1088.20024 · doi:10.1007/s10958-006-0207-6
[63] N. Gordeev 2006 Sums of orbits of algebraic groups. I J. Algebra295 1 62-80 · Zbl 1088.20022 · doi:10.1016/j.jalgebra.2005.05.006
[64] N. Gordeev 2015 On Engel words on simple algebraic groups J. Algebra425 215-244 · Zbl 1314.20038 · doi:10.1016/j.jalgebra.2014.10.045
[65] Н. Л. Гордеев, Б. Э. Кунявский, Е. Б. Плоткин 2016 Вербальные отображения и вербальные отображения с константами простых алгебраических групп Докл. РАН471 2 136-138 · doi:10.7868/S0869565216320049
[66] English transl. N. L. Gordeev, B. E. Kunyavskii, and E. B. Plotkin 2016 Word maps and word maps with constants of simple algebraic groups Dokl. Math.94 3 632-634 · Zbl 1371.20038 · doi:10.1134/s1064562416060077
[67] N. Gordeev, B. Kunyavskii, and E. Plotkin 2018 Word maps, word maps with constants and representation varieties of one-relator groups J. Algebra500 390-424 · Zbl 1401.20051 · doi:10.1016/j.jalgebra.2017.03.016
[68] N. Gordeev, B. Kunyavskii, and E. Plotkin Word maps on perfect algebraic groups Internat. J. Algebra Comput. · Zbl 1371.20038
[69] N. Gordeev, B. Kunyavskii, and E. Plotkin 2018 1801.00381 20 pp.
[70] N. Gordeev and U. Rehmann 2001 On multicommutators for simple algebraic groups J. Algebra245 1 275-296 · Zbl 0994.20039 · doi:10.1006/jabr.2001.8929
[71] N. Gordeev and J. Saxl 2005 Products of conjugacy classes in Chevalley groups over local rings Алгебра и анализ17 2 96-107
[72] N. Gordeev and J. Saxl 2006 St. Petersburg Math. J.17 2 285-293 · Zbl 1102.20033 · doi:10.1090/S1061-0022-06-00904-6
[73] S. R. Gordon 1974 Associators in simple algebras Pacific J. Math.51 131-141 · Zbl 0348.17010 · doi:10.2140/pjm.1974.51.131
[74] M. Goto and F. D. Grosshans 1978 Semisimple Lie algebras Lecture Notes Pure Appl. Math. 38 Marcel Dekker, Inc., New York-Basel vii+480 pp.
[75] M. Gromov 1987 Hyperbolic groups Essays in group theory Math. Sci. Res. Inst. Publ. 8 Springer, New York 75-263 · Zbl 0634.20015 · doi:10.1007/978-1-4613-9586-7_3
[76] M. Gromov 2003 Random walk in random groups Geom. Funct. Anal.13 1 73-146 · Zbl 1122.20021 · doi:10.1007/s000390300002
[77] R. M. Guralnick and W. M. Kantor 2000 Probabilistic generation of finite simple groups J. Algebra234 2 743-792 · Zbl 0973.20012 · doi:10.1006/jabr.2000.8357
[78] R. M. Guralnick, M. W. Liebeck, E. A. O’Brien, A. Shalev, and P. H. Tiep 2018 Surjective word maps and Burnside’s \(p^aq^b\)-theorem Invent. Math.213 2 589-695 · Zbl 1397.20037 · doi:10.1007/s00222-018-0795-z
[79] R. Guralnick and P. Shumyatsky 2015 On rational and concise words J. Algebra429 213-217 · Zbl 1346.20042 · doi:10.1016/j.jalgebra.2015.02.003
[80] R. M. Guralnick and P. H. Tiep 2015 Effective results on the Waring problem for finite simple groups Amer. J. Math.137 5 1401-1430 · Zbl 1338.20009 · doi:10.1353/ajm.2015.0035
[81] S. Jambor, M. W. Liebeck, and E. A. O’Brien 2013 Some word maps that are non-surjective on infinitely many finite simple groups Bull. Lond. Math. Soc.45 5 907-910 · Zbl 1292.20014 · doi:10.1112/blms/bdt010
[82] M. Jarden and A. Lubotzky 1999 Random normal subgroups of free profinite groups J. Group Theory2 2 213-224 · Zbl 0928.20027 · doi:10.1515/jgth.1999.015
[83] B. Hall 2015 Lie groups, Lie algebras, and representations. An elementary introduction Grad. Texts in Math. 222 Springer, Cham 2nd ed., xiv+449 pp. · Zbl 1316.22001 · doi:10.1007/978-3-319-13467-3
[84] A. Hatcher 2002 Algebraic topology Cambridge Univ. Press, Cambridge xii+544 pp.
[85] R. Hirschbühl 1990 Commutators in classical Lie algebras Linear Algebra Appl.142 91-111 · Zbl 0714.17006 · doi:10.1016/0024-3795(90)90258-E
[86] G. P. Hochschild 1981 Basic theory of algebraic groups and Lie algebras Graduate Texts in Math. 75 Springer-Verlag, New York-Berlin viii+267 pp. · doi:10.1007/978-1-4613-8114-3
[87] K. H. Hofmann and J. D. Lawson 1983 Divisible subsemigroups of Lie groups J. London Math. Soc. (2)27 3 427-434 · Zbl 0494.22002 · doi:10.1112/jlms/s2-27.3.427
[88] K. H. Hofmann and S. A. Morris 2007 The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups EMS Tracts Math. 2 Eur. Math. Soc., Zürich xvi+678 pp. · Zbl 1153.22006 · doi:10.4171/032
[89] K. H. Hofmann and W. A. F. Ruppert 1997 Lie groups and subsemigroups with surjective exponential function Mem. Amer. Math. Soc. 130 Amer. Math. Soc., Providence, RI 618 viii+174 pp. · Zbl 0889.22001 · doi:10.1090/memo/0618
[90] C. Y. Hui, M. Larsen, and A. Shalev 2015 The Waring problem for Lie groups and Chevalley groups Israel J. Math.210 1 81-100 · Zbl 1343.22006 · doi:10.1007/s11856-015-1246-9
[91] J. E. Humphreys 1995 Conjugacy classes in semisimple algebraic groups Math. Surveys Monogr. 43 Amer. Math. Soc., Providence, RI xviii+196 pp.
[92] A. Jaikin-Zapirain 2008 On the verbal width of finitely generated pro-\(p\) groups Rev. Mat. Iberoam.24 2 617-630 · Zbl 1158.20012 · doi:10.4171/RMI/549
[93] R. Kadison, Z. Liu, and A. Thom A note on commutators in algebras of unbounded operators https://tu-dresden.de/mn/math/geometrie/thom/forschung/publikationen preprint
[94] V. Kaftal, P. W. Ng, and S. Zhang 2014 Commutators and linear spans of projections in certain finite \(C^*\)-algebras J. Funct. Anal.266 4 1883-1912 · Zbl 1297.46038 · doi:10.1016/j.jfa.2013.12.009
[95] A. Kanel-Belov, B. Kunyavskiĭ, and E. Plotkin 2013 Word equations in simple groups and polynomial equations in simple algebras Вестн. С.-Петербург. ун-та. Сер.1. Матем. Мех. Астрон. 1 10-24 · Zbl 1300.20033
[96] A. Kanel-Belov, B. Kunyavskiĭ, and E. Plotkin 2013 Vestnik St. Petersburg Univ. Math.46 1 3-13 · Zbl 1300.20033 · doi:10.3103/S1063454113010044
[97] A. Kanel-Belov, S. Malev, and L. Rowen 2012 The images of non-commutative polynomials evaluated on \(2\times 2\) matrices Proc. Amer. Math. Soc.140 2 465-478 · Zbl 1241.16017 · doi:10.1090/S0002-9939-2011-10963-8
[98] A. Kanel-Belov, S. Malev, and L. Rowen 2016 The images of multilinear polynomials evaluated on \(3\times 3\) matrices Proc. Amer. Math. Soc.144 1 7-19 · Zbl 1373.16039 · doi:10.1090/proc/12478
[99] A. Kanel-Belov, S. Malev, and L. Rowen 2016 Power-central polynomials on matrices J. Pure Appl. Algebra220 6 2164-2176 · Zbl 1341.16019 · doi:10.1016/j.jpaa.2015.11.001
[100] A. Kanel-Belov, S. Malev, and L. Rowen 2017 The images of Lie polynomials evaluated on matrices Comm. Algebra45 11 4801-4808 · Zbl 1414.17005 · doi:10.1080/00927872.2017.1282959
[101] I. Kapovich and P. Schupp 2008 On group-theoretic models of randomness and genericity Groups Geom. Dyn.2 3 383-404 · Zbl 1239.20047 · doi:10.4171/GGD/45
[102] I. Kapovich and P. E. Schupp 2009 Random quotients of the modular group are rigid and essentially incompressible J. Reine Angew. Math.2009 628 91-119 · Zbl 1167.22009 · doi:10.1515/CRELLE.2009.019
[103] M. Kassabov and N. Nikolov 2013 Words with few values in finite simple groups Quart. J. Math.64 4 1161-1166 · Zbl 1296.20037 · doi:10.1093/qmath/has018
[104] E. Klimenko, B. Kunyavskiĭ, J. Morita, and E. Plotkin 2015 Word maps in Kac-Moody setting Toyama Math. J.37 25-54 · Zbl 1391.20027
[105] A. Klyachko and A. Thom 2017 New topological methods to solve equations over groups Algebr. Geom. Topol.17 1 331-353 · Zbl 1390.22006 · doi:10.2140/agt.2017.17.331
[106] A. Kulshrestha and A. Singh 2017 Computing \(n^{\text{th}}\) roots in \(SL_2(k)\) and Fibonacci polynomials 1710.03432 31 pp.
[107] B. Kunyavskiĭ 2013 Local-global invariants of finite and infinite groups: around Burnside from another side Expo. Math.31 3 256-273 · Zbl 1304.20049 · doi:10.1016/j.exmath.2013.06.004
[108] B. Kunyavskiĭ 2015 Equations in matrix groups and algebras over number fields and rings: prolegomena to a lowbrow noncommutative Diophantine geometry Arithmetic and geometry London Math. Soc. Lecture Note Ser. 420 Cambridge Univ. Press, Cambridge 264-282 · Zbl 1356.20024 · doi:10.1017/CBO9781316106877.016
[109] F. Lalonde and A. Teleman 2013 The \(g\)-areas and the commutator length Internat. J. Math.24 7 1350057 13 pp. · Zbl 1276.53082 · doi:10.1142/S0129167X13500572
[110] M. Larsen 2004 Word maps have large image Israel J. Math.139 149-156 · Zbl 1130.20310 · doi:10.1007/BF02787545
[111] M. J. Larsen and R. Pink 2011 Finite subgroups of algebraic groups J. Amer. Math. Soc.24 4 1105-1158 · Zbl 1241.20054 · doi:10.1090/S0894-0347-2011-00695-4
[112] M. Larsen and A. Shalev 2009 Word maps and Waring type problems J. Amer. Math. Soc.22 2 437-466 · Zbl 1206.20014 · doi:10.1090/S0894-0347-08-00615-2
[113] M. Larsen and A. Shalev 2012 Fibers of word maps and some applications J. Algebra354 1 36-48 · Zbl 1258.20011 · doi:10.1016/j.jalgebra.2011.10.040
[114] M. Larsen and A. Shalev 2016 On the distribution of values of certain word maps Trans. Amer. Math. Soc.368 3 1647-1661 · Zbl 1347.20081 · doi:10.1090/tran/6389
[115] M. Larsen and A. Shalev 2018 Words, Hausdorff dimension and randomly free groups Math. Ann.371 3-4 1409-1427 · Zbl 1498.20074 · doi:10.1007/s00208-017-1635-y
[116] M. Larsen, A. Shalev, and P. H. Tiep 2011 The Waring problem for finite simple groups Ann. of Math. (2)174 3 1885-1950 · Zbl 1283.20008 · doi:10.4007/annals.2011.174.3.10
[117] M. Larsen, A. Shalev, and P. H. Tiep 2013 Waring problem for finite quasisimple groups Int. Math. Res. Not. IMRN2013 10 2323-2348 · Zbl 1329.20014 · doi:10.1093/imrn/rns109
[118] M. Larsen, A. Shalev, and P. H. Tiep 2018 Probabilistic Waring problems for finite simple groups 1808.05116 44 pp.
[119] M. Larsen and P. H. Tiep 2015 A refined Waring problem for finite simple groups Forum Math. Sigma3 e6 22 pp. · Zbl 1328.20023 · doi:10.1017/fms.2015.4
[120] A. Lev 1993 Products of cyclic conjugacy classes in the groups \(\operatorname{PSL}(n,F)\) Linear Algebra Appl.179 59-83 · Zbl 0803.20027 · doi:10.1016/0024-3795(93)90321-E
[121] M. Levy 2012 Word maps with small image in simple groups 1206.1206 5 pp.
[122] M. Levy 2014 Images of word maps in almost simple groups and quasisimple groups Internat. J. Algebra Comput.24 1 47-58 · Zbl 1336.20013 · doi:10.1142/S0218196714500040
[123] M. W. Liebeck, E. A. O’Brien, A. Shalev, and P. H. Tiep 2010 The Ore conjecture J. Eur. Math. Soc. (JEMS)12 4 939-1008 · Zbl 1205.20011 · doi:10.4171/JEMS/220
[124] M. W. Liebeck, E. A. O’Brien, A. Shalev, and P. H. Tiep 2011 Commutators in finite quasisimple groups Bull. Lond. Math. Soc.43 6 1079-1092 · Zbl 1236.20011 · doi:10.1112/blms/bdr043
[125] Y. Liu and M. M. Wood 2017 The free group on \(n\) generators modulo \(n+u\) random relations as \(n\) goes to infinity 1708.08509 40 pp.
[126] A. Lonjou 2016 Non simplicité du groupe de Cremona sur tout corps Ann. Inst. Fourier (Grenoble)66 5 2021-2046 · Zbl 1365.14017 · doi:10.5802/aif.3056
[127] Z. Lu 2018 Flatness of the commutator map on special linear groups 1807.07300 32 pp.
[128] A. Lubotzky 2014 Images of word maps in finite simple groups Glasg. Math. J.56 2 465-469 · Zbl 1318.20014 · doi:10.1017/S0017089513000396
[129] A. Lubotzky and A. R. Magid 1985 Varieties of representations of finitely generated groups Mem. Amer. Math. Soc. 58 Amer. Math. Soc., Providence, RI No. 336, xi+117 pp. · Zbl 0598.14042 · doi:10.1090/memo/0336
[130] A. Ma and J. Oliva 2016 On the images of Jordan polynomials evaluated over symmetric matrices Linear Algebra Appl.492 13-25 · Zbl 1395.17075 · doi:10.1016/j.laa.2015.11.015
[131] W. Magnus 1934 Über den Beweis des Hauptidealsatzes J. Reine Angew. Math.1934 170 235-242 · JFM 60.0086.03 · doi:10.1515/crll.1934.170.235
[132] S. Malev 2014 The images of non-commutative polynomials evaluated on \(2\times 2\) matrices over an arbitrary field J. Algebra Appl.13 6 1450004 12 pp. · Zbl 1300.16022 · doi:10.1142/S0219498814500042
[133] J. Malkoun and N. Nahlus 2017 Commutators and Cartan subalgebras in Lie algebras of compact semisimple Lie groups J. Lie Theory27 4 1027-1032 · Zbl 1383.22010
[134] G. Malle 2014 The proof of Ore’s conjecture (after Ellers-Gordeev and Liebeck-O’Brien-Shalev-Tiep) Séminaire Bourbaki, v. 2012/2013 Astérisque 361 Soc. Math. France, Paris 325-348 Exp. No. 1069 · Zbl 1356.20008
[135] 2013 MathOverflow discussion: Birational automorphisms and infinite divisibility http://mathoverflow.net/questions/120818
[136] 2013 MathOverflow discussion: Word evaluating to a group element and its inverse with different frequency http://mathoverflow.net/questions/137753
[137] 2015 MathOverflow discussion: Jacobson-Morozov theorem http://mathoverflow.net/questions/210728
[138] M. McCrudden 1981 On \(n\) th roots and infinitely divisible elements in a connected Lie group Math. Proc. Cambridge Philos. Soc.89 2 293-299 · Zbl 0454.22009 · doi:10.1017/S0305004100058175
[139] G. J. McNinch 2003 Sub-principal homomorphisms in positive characteristic Math. Z.244 2 433-455 · Zbl 1036.20042 · doi:10.1007/s00209-003-0508-0
[140] G. D. Mostow 1956 Fully reducible subgroups of algebraic groups Amer. J. Math.78 200-221 · Zbl 0073.01603 · doi:10.2307/2372490
[141] A. Muranov 2007 Finitely generated infinite simple groups of infinite commutator width Internat. J. Algebra Comput.17 3 607-659 · Zbl 1141.20022 · doi:10.1142/S0218196707003767
[142] A. Myasnikov and A. Nikolaev 2014 Verbal subgroups of hyperbolic groups have infinite width J. London Math. Soc. (2)90 2 573-591 · Zbl 1339.20036 · doi:10.1112/jlms/jdu034
[143] R. K. Nath 2014 A new class of almost measure preserving maps on finite simple groups J. Algebra Appl.13 4 1350142 5 pp. · Zbl 1308.20013 · doi:10.1142/S0219498813501429
[144] B. H. Neumann 1943 Adjunction of elements to groups J. London Math. Soc.18 4-11 · Zbl 0028.33902 · doi:10.1112/jlms/s1-18.1.4
[145] P. W. Ng 2012 Commutators in the Jiang-Su algebra Internat. J. Math.23 11 1250113 29 pp. · Zbl 1267.46068 · doi:10.1142/S0129167X12501133
[146] N. Nikolov and L. Pyber 2011 Product decompositions of quasirandom groups and a Jordan type theorem J. Eur. Math. Soc. (JEMS)13 4 1063-1077 · Zbl 1228.20020 · doi:10.4171/JEMS/275
[147] Y. Ollivier 2005 A January 2005 invitation to random groups Ensaios Mat. 10 Soc. Brasil. Mat., Rio de Janeiro ii+100 pp.
[148] A. Yu. Ol’shanskiĭ 1992 Almost every group is hyperbolic Internat. J. Algebra Comput.2 1 1-17 · Zbl 0779.20016 · doi:10.1142/S0218196792000025
[149] O. Parzanchevski and G. Schul 2014 On the Fourier expansion of word maps Bull. Lond. Math. Soc.46 1 91-102 · Zbl 1322.20015 · doi:10.1112/blms/bdt068
[150] S. Pasiencier and H.-C. Wang 1962 Commutators in a semi-simple Lie group Proc. Amer. Math. Soc.13 6 907-913 · Zbl 0112.02506 · doi:10.1090/S0002-9939-1962-0169947-4
[151] В. П. Платонов, А. С. Рапинчук 1991 Алгебраические группы и теория чисел Наука, М. 656 pp.
[152] English transl. V. P. Platonov and A. S. Rapinchuk 1994 Algebraic groups and number theory Pure Appl. Math. 139 Academic Press, Inc., Boston, MA xii+614 pp.
[153] T. Plotnikov 2018 On semi-rational groups 1803.07120 8 pp.
[154] G. Prasad 1982 Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits Bull. Soc. Math. France110 2 197-202 · Zbl 0492.20029 · doi:10.24033/bsmf.1959
[155] R. Proud, J. Saxl and D. Testerman 2000 Subgroups of type \(A_1\) containing a fixed unipotent element in an algebraic group J. Algebra231 1 53-66 · Zbl 0981.20035 · doi:10.1006/jabr.2000.8353
[156] R. Ree 1964 Commutators in semi-simple algebraic groups Proc. Amer. Math. Soc.15 3 457-460 · Zbl 0127.25605 · doi:10.1090/S0002-9939-1964-0161944-X
[157] C. Riehm 1970 The norm \(1\) group of a \(p\)-adic division algebra Amer. J. Math.92 2 499-523 · Zbl 0199.37601 · doi:10.2307/2373336
[158] V. Roman’kov 2012 Equations over groups Groups Complex. Cryptol.4 2 191-239 · Zbl 1304.20058 · doi:10.1515/gcc-2012-0015
[159] J. Schneider and A. Thom 2018 Word images in symmetric and unitary groups are dense 1802.09289 18 pp.
[160] D. Segal 2009 Words: notes on verbal width in groups London Math. Soc. Lecture Note Ser. 361 Cambridge Univ. Press, Cambridge xii+121 pp. · doi:10.1017/CBO9781139107082
[161] J.-P. Serre 2009 A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field Mosc. Math. J.9 1 183-198 · Zbl 1203.14017
[162] J.-P. Serre 2010 Le groupe de Cremona et ses sous-groupes finis Séminaire Bourbaki Astérisque 332, 2008/2009 Soc. Math. France, Paris 75-100 Exp. No. 1000 · Zbl 1257.14012
[163] A. Shalev 2007 Commutators, words, conjugacy classes and character methods Turkish J. Math.31 131-148 suppl. · Zbl 1162.20014
[164] A. Shalev 2012 Applications of some zeta functions in group theory Zeta functions in algebra and geometry Contemp. Math. 566 Amer. Math. Soc., Providence, RI 331-344 · Zbl 1260.20022 · doi:10.1090/conm/566/11227
[165] A. Shalev 2013 Some results and problems in the theory of word maps Erdős centennial Bolyai Soc. Math. Stud. 25 János Bolyai Math. Soc., Budapest 611-649 · Zbl 1321.20032 · doi:10.1007/978-3-642-39286-3_22
[166] M. Slusky 2010 Zeros of \(2\times 2\) matrix polynomials Comm. Algebra38 11 4212-4223 · Zbl 1227.15016 · doi:10.1080/00927870903366843
[167] Š. Špenko 2013 On the image of a noncommutative polynomial J. Algebra377 298-311 · Zbl 1292.16016 · doi:10.1016/j.jalgebra.2012.12.006
[168] T. A. Springer and R. Steinberg 1970 Conjugacy classes Seminar on algebraic groups and related finite groupsThe Institute for Advanced Study, Princeton, NJ 1968/69 Lecture Notes in Math. 131 Springer, Berlin 167-266 · doi:10.1007/BFb0081546
[169] A. Stein \(1998 1\frac{1}{2} \)-generation of finite simple groups Beiträge Algebra Geom.39 2 349-358 · Zbl 0924.20027
[170] R. Steinberg 1965 Regular elements of semisimple algebraic groups Inst. Hautes Études Sci. Publ. Math.25 49-80 · Zbl 0136.30002 · doi:10.1007/BF02684397
[171] R. Steinberg 2016 Lectures on Chevalley groups, Lectures notes, Yale Univ., 1967-1968 Univ. Lecture Series 66 Amer. Math. Soc., Providence, RI 2nd ed., xi+160 pp. · Zbl 1361.20003 · doi:10.1090/ulect/066
[172] R. Steinberg 1974 Conjugacy classes in algebraic groups Lecture Notes in Math. 366 Springer-Verlag, Berlin-New York vi+159 pp. · doi:10.1007/BFb0067854
[173] R. Steinberg 2003 On power maps in algebraic groups Math. Res. Lett.10 5-6 621-624 · Zbl 1045.20044 · doi:10.4310/MRL.2003.v10.n5.a5
[174] D. M. Testerman \(1995 A_1\)-type overgroups of elements of order \(p\) in semisimple algebraic groups and the associated finite groups J. Algebra177 1 34-76 · Zbl 0857.20025 · doi:10.1006/jabr.1995.1285
[175] A. Thom 2013 Convergent sequences in discrete groups Canad. Math. Bull.56 2 424-433 · Zbl 1276.54028 · doi:10.4153/CMB-2011-155-3
[176] T. Tsuboi 2012 On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds Comment. Math. Helv.87 1 141-185 · Zbl 1236.57039 · doi:10.4171/CMH/251
[177] T. Tsuboi 2013 Homeomorphism groups of commutator width one Proc. Amer. Math. Soc.141 5 1839-1847 · Zbl 1279.54023 · doi:10.1090/S0002-9939-2012-11595-3
[178] L. Vaserstein and E. Wheland 1995 Products of conjugacy classes of two by two matrices Linear Algebra Appl.230 165-188 · Zbl 0840.20040 · doi:10.1016/0024-3795(94)00005-X
[179] Э. Б. Винберг, А. Л. Онищик 1995 Семинар по группам Ли и алгебраическим группам УРСС, М. 2-е изд.344 pp.
[180] English transl. of 1st ed. A. L. Onishchik and E. B. Vinberg 1990 Lie groups and algebraic groups Springer Ser. Soviet Math. Springer-Verlag, Berlin xx+328 pp. · doi:10.1007/978-3-642-74334-4
[181] B. A. F. Wehrfritz 1969 A residual property of free metabelian groups Arch. Math. (Basel)20 3 248-250 · Zbl 0184.04201 · doi:10.1007/BF01899293
[182] J. W. Wood 1971 Bundles with totally disconnected structure group Comment. Math. Helv.46 257-273 · Zbl 0217.49202 · doi:10.1007/BF02566843
[183] M. Wüstner 2002 Historical remarks on the surjectivity of the exponential function of Lie groups Historia Math.29 3 266-272 · Zbl 1012.22008 · doi:10.1006/hmat.2002.2357
[184] M. Wüstner 2005 The classification of all simple Lie groups with surjective exponential map J. Lie Theory15 1 269-278 · Zbl 1072.22003
[185] S. Zimmermann 2018 The Abelianization of the real Cremona group Duke Math. J.167 2 211-267 · Zbl 1402.14015 · doi:10.1215/00127094-2017-0028
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