Approximations of groups, characterizations of sofic groups, and equations over groups. (English) Zbl 1364.20013
Author’s abstract: We give new characterizations of sofic groups:
- –
- A group \(G\) is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups.
- –
- A group \(G\) is sofic if and only if any system of equations solvable in all alternating groups is solvable over \(G\).
Reviewer: Alexander Ivanovich Budkin (Barnaul)
MSC:
| 20E26 | Residual properties and generalizations; residually finite groups |
| 20E18 | Limits, profinite groups |
| 20F70 | Algebraic geometry over groups; equations over groups |
References:
| [1] | Gromov, M., Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS), 1, 2, 109-197 (1999) · Zbl 0998.14001 |
| [2] | Weiss, B., Sofic groups and dynamical systems, Sankhya, Ser. A, 62, 3, 350-359 (2000), Ergodic Theory and Harmonic Analysis (Mumbai, 1999) · Zbl 1148.37302 |
| [3] | Alekseev, M. A.; Glebskiĭ, L. Y.; Gordon, E. I., On approximations of groups, group actions and Hopf algebras, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 256, Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 3, 224-262 (1999), 268 · Zbl 1130.20306 |
| [4] | Rădulescu, F., The von Neumann algebra of the non-residually finite Baumslag group \(\langle a, b | a b^3 a^{- 1} = b^2 \rangle\) embeds into \(R^\omega \), (Hot Topics in Operator Theory. Hot Topics in Operator Theory, Theta Ser. Adv. Math., vol. 9 (2008), Theta: Theta Bucharest), 173-185 · Zbl 1199.46137 |
| [5] | Capraro, V.; Lupini, M., Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture, Lecture Notes in Mathematics, vol. 2136 (2015), Springer: Springer Cham, with an appendix by Vladimir Pestov · Zbl 1383.20002 |
| [6] | Pestov, V. G., Hyperlinear and sofic groups: a brief guide, Bull. Symbolic Logic, 14, 4, 449-480 (2008) · Zbl 1206.20048 |
| [7] | Elek, G.; Szabó, E., Hyperlinearity, essentially free actions and \(L^2\)-invariants. The sofic property, Math. Ann., 332, 2, 421-441 (2005) · Zbl 1070.43002 |
| [8] | Thom, A., About the metric approximation of Higman’s group, J. Group Theory, 15, 2, 301-310 (2012) · Zbl 1251.20040 |
| [9] | Arzhantseva, G. N., Asymptotic approximations of finitely generated groups, (González-Meneses, J.; Lustig, M.; Ventura, E., Extended Abstracts Fall 2012. Extended Abstracts Fall 2012, Research Perspectives CRM Barcelona, Trends in Mathematics, vol. 1 (2014), Springer), 7-16 |
| [10] | Glebsky, L.; Rivera, L. M., Sofic groups and profinite topology on free groups, J. Algebra, 320, 9, 3512-3518 (2008) · Zbl 1162.20015 |
| [11] | Arzhantseva, G. N.; Paunescu, L., Linear sofic groups and algebras, Trans. Amer. Math. Soc., 369, 2285-2310 (2017) · Zbl 1406.20038 |
| [12] | Kharlampovich, O.; Myasnikov, A., Elementary theory of free non-abelian groups, J. Algebra, 302, 2, 451-552 (2006) · Zbl 1110.03020 |
| [13] | Sela, Z., Diophantine geometry over groups. VI. The elementary theory of a free group, Geom. Funct. Anal., 16, 3, 707-730 (2006) · Zbl 1118.20035 |
| [14] | Coulbois, T.; Khelif, A., Equations in free groups are not finitely approximable, Proc. Amer. Math. Soc., 127, 4, 963-965 (1999) · Zbl 0914.20031 |
| [16] | Higman, G.; Scott, E., Existentially Closed Groups, London Mathematical Society Monographs, New Series, vol. 3 (1988), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, Oxford Science Publications · Zbl 0646.20001 |
| [17] | Stallings, J. R., Surfaces in three-manifolds and nonsingular equations in groups, Math. Z., 184, 1, 1-17 (1983) · Zbl 0496.57006 |
| [18] | Brick, S. G., Normal-convexity and equations over groups, Invent. Math., 94, 1, 81-104 (1988) · Zbl 0655.20025 |
| [19] | Brick, S. G., Relative normal-convexity and amalgamations, Bull. Aust. Math. Soc., 44, 1, 95-107 (1991) · Zbl 0725.20019 |
| [21] | Howie, J., The \(p\)-adic topology on a free group: a counterexample, Math. Z., 187, 1, 25-27 (1984) · Zbl 0548.20017 |
| [22] | Liebeck, M. W.; Shalev, A., Simple groups, permutation groups, and probability, J. Amer. Math. Soc., 12, 2, 497-520 (1999) · Zbl 0916.20003 |
| [23] | Brenner, J. L., Covering theorems for FINASIGs. VIII. Almost all conjugacy classes in \(A_n\) have exponent ≤4, J. Aust. Math. Soc. A, 25, 2, 210-214 (1978) · Zbl 0374.20039 |
| [24] | Baumslag, G.; Myasnikov, A.; Remeslennikov, V., Algebraic geometry over groups. I. Algebraic sets and ideal theory, J. Algebra, 219, 1, 16-79 (1999) · Zbl 0938.20020 |
| [25] | Brown, N. P.; Ozawa, N., \(C^\ast \)-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics, vol. 88 (2008), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1160.46001 |
| [26] | Ol’shanskiĭ, A. Y., SQ-universality of hyperbolic groups, Mat. Sb., 186, 8, 119-132 (1995) · Zbl 0864.20023 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
