New topological methods to solve equations over groups

Klyachko, Anton¹ ; Thom, Andreas²

¹Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia
²Institut für Geometrie, TU Dresden, D-01062 Dresden, Germany

Algebraic and Geometric Topology, Tome 17 (2017) no. 1, pp. 331-353

Résumé

We show that the equation associated with a group word w ∈ G ∗ F2 can be solved over a hyperlinear group G if its content — that is, its augmentation in F2 — does not lie in the second term of the lower central series of F2. Moreover, if G is finite, then a solution can be found in a finite extension of G. The method of proof extends techniques developed by Gerstenhaber and Rothaus, and uses computations in p–local homotopy theory and cohomology of compact Lie groups.

Détail

DOI : 10.2140/agt.2017.17.331

Classification : 22C05, 20F70
Keywords: equations over groups, cohomology of Lie groups

Bibliographie

[1] D V Baranov, A A Klyachko, Economical adjunction of square roots to groups, Sibirsk. Mat. Zh. 53 (2012) 250

[2] P F Baum, W Browder, The cohomology of quotients of classical groups, Topology 3 (1965) 305 | DOI

[3] A Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. 57 (1953) 115 | DOI

[4] A Borel, Sur l’homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math. 76 (1954) 273 | DOI

[5] R Bott, The space of loops on a Lie group, Michigan Math. J. 5 (1958) 35 | DOI

[6] R Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960) 249 | DOI

[7] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972) | DOI

[8] M Edjvet, J Howie, The solution of length four equations over groups, Trans. Amer. Math. Soc. 326 (1991) 345 | DOI

[9] M Edjvet, A Juhász, Nonsingular equations over groups, II, Comm. Algebra 38 (2010) 1640 | DOI

[10] M Edjvet, A Juhász, Non-singular equations over groups, I, Algebra Colloq. 18 (2011) 221 | DOI

[11] A Elkasapy, A Thom, About Gotô’s method showing surjectivity of word maps, Indiana Univ. Math. J. 63 (2014) 1553 | DOI

[12] A Evangelidou, The solution of length five equations over groups, Comm. Algebra 35 (2007) 1914 | DOI

[13] S M Gersten, Reducible diagrams and equations over groups, from: "Essays in group theory" (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 15 | DOI

[14] M Gerstenhaber, O S Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. USA 48 (1962) 1531 | DOI

[15] M Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. 1 (1999) 109 | DOI

[16] H Hamanaka, D Kishimoto, A Kono, Self homotopy groups with large nilpotency classes, Topology Appl. 153 (2006) 2425 | DOI

[17] A Hatcher, Algebraic topology, Cambridge University Press (2002)

[18] G Higman, A finitely generated infinite simple group, J. London Math. Soc. 26 (1951) 61 | DOI

[19] H Hopf, Über den Rang geschlossener Liescher Gruppen, Comment. Math. Helv. 13 (1940) 119 | DOI

[20] J Howie, On pairs of 2–complexes and systems of equations over groups, J. Reine Angew. Math. 324 (1981) 165 | DOI

[21] J Howie, The solution of length three equations over groups, Proc. Edinburgh Math. Soc. 26 (1983) 89 | DOI

[22] S V Ivanov, A A Klyachko, Solving equations of length at most six over torsion-free groups, J. Group Theory 3 (2000) 329 | DOI

[23] I James, E Thomas, Which Lie groups are homotopy-abelian ?, Proc. Nat. Acad. Sci. USA 45 (1959) 737 | DOI

[24] A Juhász, On the solvability of equations over groups, Comm. Algebra 32 (2004) 1487 | DOI

[25] D Kishimoto, A Kono, On a conjecture of Ōshima, Topology Appl. 156 (2009) 2189 | DOI

[26] A A Klyachko, A funny property of sphere and equations over groups, Comm. Algebra 21 (1993) 2555 | DOI

[27] A A Klyachko, Equations over groups, quasivarieties, and a residual property of a free group, J. Group Theory 2 (1999) 319 | DOI

[28] A A Klyachko, How to generalize known results on equations over groups, Mat. Zametki 79 (2006) 409

[29] A A Klyachko, A V Trofimov, The number of non-solutions of an equation in a group, J. Group Theory 8 (2005) 747 | DOI

[30] F Levin, Solutions of equations over groups, Bull. Amer. Math. Soc. 68 (1962) 603 | DOI

[31] A Malcev, On isomorphic matrix representations of infinite groups, Mat. Sbornik N.S. 8 (50) (1940) 405

[32] J P May, K Ponto, More concise algebraic topology: localization, completion, and model categories, University of Chicago Press (2012)

[33] M Mimura, G Nishida, H Toda, Localization of CW–complexes and its applications, J. Math. Soc. Japan 23 (1971) 593 | DOI

[34] B H Neumann, Adjunction of elements to groups, J. London Math. Soc. 18 (1943) 4 | DOI

[35] N Nikolov, D Segal, Generators and commutators in finite groups; abstract quotients of compact groups, Invent. Math. 190 (2012) 513 | DOI

[36] V G Pestov, Hyperlinear and sofic groups: a brief guide, Bull. Symbolic Logic 14 (2008) 449 | DOI

[37] G J Porter, Homotopical nilpotence of S3, Proc. Amer. Math. Soc. 15 (1964) 681 | DOI

[38] V K Rao, Spin(n) is not homotopy nilpotent for n ≥ 7, Topology 32 (1993) 239 | DOI

[39] V Roman’Kov, Equations over groups, Groups Complex. Cryptol. 4 (2012) 191 | DOI

[40] O S Rothaus, On the non-triviality of some group extensions given by generators and relations, Ann. of Math. 106 (1977) 599 | DOI

[41] H Samelson, Groups and spaces of loops, Comment. Math. Helv. 28 (1954) 278 | DOI

[42] J P Serre, Groupes d’homotopie et classes de groupes abéliens, Ann. of Math. 58 (1953) 258 | DOI

[43] J P Serre, Trees, Springer (2003) | DOI

[44] A Stolz, A Thom, On the lattice of normal subgroups in ultraproducts of compact simple groups, Proc. Lond. Math. Soc. 108 (2014) 73 | DOI

[45] S Stolz, P Teichner, What is an elliptic object?, from: "Topology, geometry and quantum field theory" (editor U Tillmann), London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 247 | DOI

[46] A Thom, Convergent sequences in discrete groups, Canad. Math. Bull. 56 (2013) 424 | DOI

[47] B Weiss, Sofic groups and dynamical systems, Sankhyā Ser. A 62 (2000) 350

[48] G W Whitehead, Elements of homotopy theory, 61, Springer (1978)

[49] N Yagita, Homotopy nilpotency for simply connected Lie groups, Bull. London Math. Soc. 25 (1993) 481 | DOI