Mathematica Bohemica, Vol. 147, No. 1, pp. 11-18, 2022
Sofic groups are not locally embeddable into finite Moufang loops
Heghine Ghumashyan, Jaroslav Guričan
Received April 13, 2020. Published online March 9, 2021.
Abstract: We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M. Vodička, P. Zlatoš (2019).
Keywords: group; diassociative IP loop; Moufang loop; finite embeddability property; local embeddability
References: [1] G. Baumslag, D. Solitar: Some two-generator one-relator non-Hopfian groups. Bull. Am. Math. Soc. 68 (1962), 199-201. DOI 10.1090/S0002-9904-1962-10745-9 | MR 0142635 | Zbl 0108.02702
[2] T. Ceccherini-Silberstein, M. Coornaert: Cellular Automata and Groups. Springer Monographs in Mathematics. Springer, Berlin (2010). DOI 10.1007/978-3-642-14034-1 | MR 2683112 | Zbl 1218.37004
[3] B. Collins, K. J. Dykema: Free products of sofic groups with amalgamation over monotileably amenable groups. Münster J. Math. 4 (2011), 101-118. MR 2869256 | Zbl 1242.43003
[4] A. Drápal: A simplified proof of Moufang's theorem. Proc. Am. Math. Soc. 139 (2011), 93-98. DOI 10.1090/S0002-9939-2010-10501-4 | MR 2729073 | Zbl 1215.20059
[5] L. Y. Glebsky, Y. I. Gordon: On approximation of amenable groups by finite quasigroups. J. Math. Sci. 140 (2007), 369-375. DOI 10.1007/s10958-007-0446-1 | MR 2183215 | Zbl 1159.43300
[6] L. Henkin: Two concepts from the theory of models. J. Symb. Log. 21 (1956), 28-32. DOI 10.2307/2268482 | MR 0075895 | Zbl 0071.00701
[7] G. Higman, B. H. Neumann, H. Neumann: Embedding theorems for groups. J. Lond. Math. Soc. 24 (1949), 247-254. DOI 10.1112/jlms/s1-24.4.247 | MR 0032641 | Zbl 0034.30101
[8] R. C. Lyndon, P. E. Schupp: Combinatorial Group Theory. Classics in Mathematics. Springer, Berlin (2001). DOI 10.1007/978-3-642-61896-3 | MR 1812024 | Zbl 0997.20037
[9] W. Magnus, A. Karrass, D. Solitar: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Dover Books on Mathematics. Dover Publications, Mineola (2004). MR 2109550 | Zbl 1130.20307
[10] A. I. Mal'tsev: On a general method for obtaining local theorems in group theory. Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Fak. 1 (1941), 3-9. (In Russian.) MR 0075939
[11] A. I. Mal'tsev: On homomorphisms onto finite groups. Twelve Papers in Algebra American Mathematical Society Translations: Series 2, 119. American Mathematical Society (1983), 67-79. DOI 10.1090/trans2/119 | Zbl 0511.20026
[12] S. Meskin: Nonresidually finite one-relator groups. Trans. Am. Math. Soc. 164 (1972), 105-114. DOI 10.2307/1995962 | MR 285589 | Zbl 0245.20028
[13] H. O. Pflugfelder: Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics 7. Heldermann Verlag, Berlin (1990). MR 1125767 | Zbl 0715.20043
[14] A. M. Vershik, E. I. Gordon: Groups that are locally embeddable in the class of finite groups. St. Petersbg. Math. J. 9 (1998), 49-67. MR 1458419 | Zbl 0898.20016
[15] M. Vodička, P. Zlatoš: The finite embeddability property for IP loops and local embeddability of groups into finite IP loops. Ars Math. Contemp. 17 (2019), 535-554. DOI 10.26493/1855-3974.1884.5cb | MR 4041359 | Zbl 1442.20040
[16] B. Weiss: Monotileable amenable groups. Topology, Ergodic Theory, Real Algebraic Geometry: Rokhlin's Memorial American Mathematical Society Translations: Series 2, 202. American Mathematical Society (2001), 257-262. DOI 10.1090/trans2/202/18 | MR 1819193 | Zbl 0982.22004
[17] M. Ziman: Extensions of Latin subsquares and local embeddability of groups and group algebras. Quasigroups Relat. Syst. 11 (2004), 115-125. MR 2064165 | Zbl 1060.20057
Affiliations: Heghine Ghumashyan, Vanadzor State University, Tigran Mets Ave, Vanadzor, Armenia; Department of Algebra and Geometry, Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Mlynská dolina 6280, 842 48 Bratislava, e-mail: hgumashyan@mail.ru; Jaroslav Guričan, Department of Algebra and Geometry, Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Mlynská dolina 6280, 842 48 Bratislava, Slovakia e-mail: gurican@fmph.uniba.sk
