Algebraic geometry over groups. I: Algebraic sets and ideal theory. (English) Zbl 0938.20020

The authors present the first in the series of three papers with the object to lay the foundations of the theory of ideals and algebraic sets over groups. There is discovered a surprising similarity to elementary algebraic geometry – hence its name.
In the present paper they introduce group-theoretic counterparts to such notions as zero-divisors, prime ideals, the Lasker-Noether decomposition of ideals as intersections of prime ideals, the Noetherian condition, irreducibility, and the Nullstellensatz.
Some new concepts arise that are interesting for the group theory. The main of them is the notion of a \(G\)-group, where \(G\) is a fixed group. This group \(G\) plays the role of the coefficient ring. A group \(H\) is called a \(G\)-group if \(H\) contains a designated copy of \(G\). Such groups \(G\) form a category with naturally defined \(G\)-morphisms. The kernels of morphisms are called ideals. One can talk in a natural way about free \(G\)-groups, finitely generated and finitely presented \(G\)-groups and so on. In particular, the finitely generated free \(G\)-groups are the free products of \(G\) with the free groups of finite ranks. So, we can consider such a group as a non-commutative analogue of a polynomial algebra over a unitary commutative ring in finitely many variables.
Elementary properties of algebraic sets and the Zariski topology are developed. The important notion of coordinate group is introduced. Equivalence of the categories of affine algebraic sets and coordinate groups is explained. Some decomposition theorems are proved.


20E05 Free nonabelian groups
20E34 General structure theorems for groups
14A22 Noncommutative algebraic geometry
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E07 Subgroup theorems; subgroup growth
20F65 Geometric group theory
20J15 Category of groups
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[1] Auslander, L., On a problem of philip Hall, Ann. of math. (2), 86, 112-116, (1967) · Zbl 0149.26904
[2] Bass, H., Groups acting on non-Archimedian trees, Arboreal group theory, 69-130, (1991)
[3] Baumslag, B., Residually free groups, Proc. London math. soc., 17, 402-418, (1967) · Zbl 0166.01502
[4] Baumslag, G., On generalized free products, Math. Z., 7, 423-438, (1962) · Zbl 0104.24402
[5] Baumslag, G.; Myasnikov, A.; Remeslennikov, V., Residually hyperbolic groups and approximation theorems for extensions of centralizers, Proc. inst. appl. math Russian acad. sci., 24, 3-37, (1996)
[6] G. Baumslag, A. Myasnikov, and, V. Remeslennikov, Discriminating completions of hyperbolic groups, submitted for publication. · Zbl 1011.20041
[7] Baumslag, G.; Myasnikov, A.; Roman’kov, V., Two theorems about equationally Noetherian groups, J. algebra, 194, 654-664, (1997) · Zbl 0888.20017
[8] Bryant, R., The verbal topology of a group, J. algebra, 48, 340-346, (1977) · Zbl 0408.20022
[9] Chang, C.C.; Keisler, H.J., Model theory, (1973), North-Holland New York · Zbl 0276.02032
[10] Comerford, L.P.; Edmunds, C.C., Quadratic equations over free groups and free products, J. algebra, 68, 276-297, (1981) · Zbl 0526.20024
[11] Fine, B.; Gaglione, A.M.; Miasnikov, A.; Rosenberger, G.; Spellman, D., A classification of fully residual free groups of rank three or less, J. algebra, 200, 571-605, (1998) · Zbl 0899.20009
[12] Grigorchuk, R.I.; Kurchanov, P.F., Some questions of group theory related to geometry, Itogi nauki i techniki, sovremennye problemy matematiki, fundamental’nye napravlenia, VINITI, 58, Encyclopedia of mathematical sciences, (1990) · Zbl 0781.20023
[13] Guba, V., Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Mat. zametki, 40, 321-324, (1986) · Zbl 0611.20020
[14] Hartshorne, R., Algebraic geometry, (1977), Springer-Verlag New York · Zbl 0367.14001
[15] Hrushovski, E.; Zilber, B., Zariski geometries, J. amer. math. soc., 9, 1-56, (1996) · Zbl 0843.03020
[16] Karrass, A.; Magnus, W.; Solitar, D., Elements of finite order in groups with a single defining relation, Comm. pure appl. math., 13, 458-466, (1960)
[17] Kharlampovich, O.; Myasnikov, A., Irreducible affine varieties over a free group. I: irreducibility of quadratic equations and nullstellensatz, J. algebra, 200, 472-516, (1998) · Zbl 0904.20016
[18] Kharlampovich, O.; Myasnikov, A., Irreducible affine varieties over a free group. II: systems in triangular quasi-quadratic form and description of residually free groups, J. algebra, 200, 517-570, (1988) · Zbl 0904.20017
[19] Lyndon, R.C., The equation a2b2=c2 in free groups, Michigan math. J., 6, 155-164, (1959)
[20] Lyndon, R.C., Groups with parametric exponents, Trans. amer. math. soc., 96, 518-533, (1960) · Zbl 0108.02501
[21] Lyndon, R.C., Equation in free groups, Trans. amer. math. soc., 96, 445-457, (1960) · Zbl 0108.02301
[22] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory: presentations of groups in terms of generators and relations, (1966), Wiley-Interscience New York/London/Sydney · Zbl 0138.25604
[23] Makanin, G.S., Decidability of the universal and positive theories of a free group, Izv. akad. nauk SSSR ser. math., 48, 735-749, (1985) · Zbl 0578.20001
[24] Myasnikov, A.G.; Remeslennikov, V.N., Exponential groups I: foundations of the theory and tensor completion, Siberian math. J., 35, 1106-1118, (1994) · Zbl 0851.20050
[25] Myasnikov, A.G.; Remeslennikov, V.N., Exponential groups II: extensions of centralizers and tensor completion of CSA-groups, Internat. J. algebra comput., 6, 687-711, (1996) · Zbl 0866.20014
[26] B. Plotkin, Varieties of algebras and algebraic varieties. Categories of algebraic varieties, Hebrew University, Jerusalem, 1996, preprint. · Zbl 0921.08004
[27] Razborov, A., On systems of equations in a free group, Math. USSR-izv., 25, 115-162, (1985) · Zbl 0579.20019
[28] A. Razborov, On systems of equations in a free group, Ph.D. thesis, Steklov Math. Institute, Moscow, 1987. · Zbl 0632.94030
[29] Remeslennikov, V.N., E-free groups, Siberian math. J., 30, 153-157, (1989) · Zbl 0724.20025
[30] Remeslennikov, V.N., Matrix representation of finitely generated metabelian groups, Algebra i logika, 8, 72-76, (1969)
[31] Scott, W.R., Algebraically closed groups, Proc. amer. math. soc., 2, 118-212, (1951) · Zbl 0043.02302
[32] Schutzenberger, M.P., Sur l’equation a2th=b2+mc2+p dans un group libre, C.R. acad. sci. Paris, 248, 2435-2436, (1959) · Zbl 0090.24403
[33] Stallings, J., Finiteness properties of matrix representations, Ann. of math., 124, 337-346, (1986) · Zbl 0604.20027
[34] Wehrfritz, B.A.F., Infinite linear groups, (1973), Springer-Verlag New York/Heidelberg/Berlin · Zbl 0261.20038
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