In 1997, B. I. Plotkin introduced a concept of geometric equivalence of algebraic structures and posed a question: is it true that every nilpotent torsion-free group is geometrically equivalent to its Mal’cev’s closure? A negative answer in the form of three counterexamples was given by V. V. Bludov and B. V. Gusev in 2007. In the present paper, an infinite series of counterexamples of unbounded Hirsch rank and nilpotency degree is constructed.
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References
I. Belegradek, “On co-Hopfian nilpotent groups,” Bull. London Math. Soc., 35, 805–811 (2003).
A. Berzins, “Geometrical equivalence of algebras,” Int. J. Algebra Comput., 11, No. 4, 447–456 (2001).
V. V. Bludov and B. V. Gusev, “Geometric equivalence of groups,” Tr. Inst. Mat. Mekh. UrO RAN, 13, No. 1, 57–78 (2007).
G. Baumslag, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over groups. I. Algebraic sets and ideal theory,” J. Algebra, 219, 16–79 (1999).
R. Camm, “Simple free products,” J. London Math. Soc., 28, 66–76 (1953).
R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin–Heidelberg–New York (1977).
M. Goze and Yu. B. Khakimdjanov, Nilpotent Lie Algebras, Kluwer Academics Publishers (1996).
É. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, Structure of Lie Groups and Lie Algebras, Springer-Verlag (1994).
R. W Goodman, Nilpotent Lie Groups: Structure and Applications to Analysis, Springer-Verlag, Berlin/New York (1976).
R. Göbel and S. Shelah, “Radicals and Plotkin’s problem concerning geometrically equivalent groups,” Proc. Amer. Math. Soc., 130, No. 3, 673–674 (2002).
H. Hamrouni, “Euler characteristics on a class of finitely generated nilpotent groups,” Osaka J. Math., 50, 339–346 (2013).
G. Hochschild, The Structure of Lie Groups, Holden-Day Inc., San Francisco (1965).
M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the Theory of Groups, Springer-Verlag, New York (1979).
Yu. Khakimdjanov, “Characteristically nilpotent Lie algebras,” Math. Sb., 181, No. 5, 642–655 (1990).
E. I. Khukhro and V. D. Mazurov, Unsolved Problems in Group Theory. The Kourovka Notebook, 16th ed., Novosibirsk (2006).
G. Leger and S. Tôgô, “Characteristically nilpotent Lie algebras,” Duke Math. J., 26, 623–628 (1959).
A. I. Mal’cev, “Nilpotent torsion-free groups,” Izv. Akad. Nauk. SSSR. Ser. Mat., 13, 201–212 (1949).
A. I. Mal’cev, “On a class of homogeneous spaces,” Izv. Akad. Nauk. SSSR. Ser. Mat., 13, 9–32 (1949).
A. A. Mishchenko, Model-theoretic and algebra-geometric problems for nilpotent partial commutative groups, Ph.D. thesis, Omsk (2009).
A. Myasnikov and V. Remeslennikov, “Algebraic geometry over groups II, Logical foundations,” J. Algebra, 234, No. 1, 225–276 (2000).
D. Nikolova and B. Plotkin, “Some notes on universal algebraic geometry,” in: Algebra. Proc. Int. Conf. on Algebra on the Occasion of the 90th Birthday of A. G. Kurosh, Moscow (1998), Walter De Gruyter Publ., Berlin (1999), pp. 237–261.
B. Plotkin, “Algebraic logic, varieties of algebras and algebraic varieties,” in: Proc. Int. Alg. Conf., St. Petersburg (1995), St.Petersburg (1999), pp. 189–271.
B. Plotkin, “Varieties of algebras and algebraic varieties,” Israel J. Math., 96, No. 2, 511–522 (1996).
B. Plotkin, “Some notions of algebraic geometry in universal algebra,” Algebra Analiz.9, No. 4, 224–248 (1997).
B. Plotkin, “Varieties of algebras and algebraic varieties: Categories of algebraic varieties,” Siberian Adv. Math., 7, No. 2, 64–97 (1997).
B. Plotkin, “Radicals in groups, operations on classes of groups, and radical classes,” Transl., II Ser. Amer. Math. Soc., 119, 89–118 (1983).
B. Plotkin, E. Plotkin, and A. Tsurkov, “Geometrical equivalence of groups,” Commun. Algebra, 27, No. 8, 4015–4025 (1999).
M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer (1972).
A. Tsurkov, “Geometrical equivalence of nilpotent groups,” Zap. Nauchn. Semin. POMI, 330, 259–270 (2006).
S. Yamaguchi, “On some classes of nilpotent Lie algebras and their automorphism groups,” Mem. Fac. Sci. Kyushu Univ., 35, 341–351 (1981).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 470, 2018, pp. 147–161.
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Noskov, G.A. Plotkin’s Geometric Equivalence, Mal’cev’s Closure, and Incompressible Nilpotent Groups. J Math Sci 243, 601–611 (2019). https://doi.org/10.1007/s10958-019-04561-x
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DOI: https://doi.org/10.1007/s10958-019-04561-x