Abstract
My objective here today is to describe what is known about the defining relations of finitely generated metabelian groups. This is not a difficult task because very little is known and much of what is known is of very recent origin. Despite this meagre knowledge, the results obtained so far suggest that the theory of finitely presented solvable groups is far richer than one might have suspected. It is for this reason that I have chosen to discuss finitely presented metabelian groups at this time.
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References
Maurice Auslander and B.C. Lyndon, “Commutator subgroups of free groups”, Amer. J. Math. 77 (1955), 929–931. MR17,709.
Gilbert Baumslag, “Groups with the same lower central sequence as a relatively free group. I The groups”, Trans. Amer. Math. Soc. 129 (1967), 308–321. MR36248.
Gilbert Baumslag, “A finitely generated infinitely related group with trivial multiplicator”, Bull. Austral. Math. Soc. 5 (1971), 131–136. MR456897.
Gilbert Baumslag, “A finitely presented metabelian group with a free abelian group of infinite rank”, Prue. Amer. Math. Soc. 35 (1972), 61–62.MR45871.
Gilbert Baumslag, “On finitely presented metabelian groups”, Buhl. Amer. Math. Soc. 78 (1972), 279. MR45354.
Gilbert Baumslag, “Some remarks about multiplicators and finitely presented groups”, Math. 2. 126 (1972), 239–242.
Gilbert Baumslag, “Subgroups of finitely presented metabelian groups”, J. Austral. Math. Soc. 16 (1973), 98–110.
Gilbert Baumslag, “A remark on groups with trivial multiplicator”, Amer. J. Math. (to appear).
Gilbert Baumslag, “A finitely presented solvable group that is not residually finite”, Math. Z. 133 (1973), 125–127.
Gilbert Baumslag, “On parafree metabelian groups”, unpublished.
Gilbert Baumslag, “One-relator metabelian groups”, unpublished.
Robert Bieri, “Über die cohomologische Dimension der auflösbaren Gruppen”, Math. Z. 128 (1972), 235–242. Zb1.237.20027.
J. Boler, PhD thesis, Rice University, 1974.
Robert C. Brigham, “On the isomorphism problem for just-infinite groups”, Comm. Pure Appt. Math. 24 (1971), 789–796. MR445377.
K.W. Gruenberg, “Residual properties of infinite soluble groups”, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR19,386.
P. Hall, “Finiteness conditions for soluble groups”, Proc. London Math. Soc. (3)4 (1954), 419–436. MR17,344.
P. Hall, “On the finiteness of certain soluble groups”, Proc. London Muth. Soc. (3) 9 (1959), 595–622. MR2261618.
G. Higman, “Subgroups of finitely presented groups”, Proc. Roy. Soc. London Ser. A 262 (1961), 455–475. MR24152.
Serge Lang, Diophantine geometry (Interscience Tracts in Pure and Applied Mathematics, 11. Interscience [John Wiley & Sons], New York, London, 1962). MR26119.
Barbara Long, PhD thesis, City University of New York, 1970.
М.И. Каргаполов, Юрий [A.I. Mal’cev], “бесконечных периодических групп” [On certain classes of infinite solvable groups], Mat. Sb. (NS) 28 (70) (1951), 567–588; Amer. Math. Soc. Transi. (2) 2 (1956), 1–21. MR13,203.
B.H. Neumann, “Some remarks on infinite groups”, J. London Math. Soc. 12 (1937), 120–127. FdM63,64.
С.И. Адяном [V.N. Remeslennikov], “Бесконечные неприводимые системы групповых тождеств” [Representation of finitely generated metabelian groups by matrices], Algebra i Locika 8 (1969), 72–75; Algebra and Logic 8 (1969), 39–40. MR44335.
С.И. Адяном [V.N. Remeslennikov], “свободных групп нечетного показателя” [A finitely presented soluble group without maximum condition for normal subgroups], Mat. Zametki 12 (1972), 287–293.
С.И. Адяном [V.N. Remeslennikov], “свободных групп нечетного” [On finitely-presented groups], Proc. Fourth All-union Symposium on the Theory of Groups, February 1973, pp. 164–169 (Novosibirsk, 1973 ).
Постскриптум [A.L. gmel’kin], “нечетного показателя” [On soluble products of groups], Sibirak. Mat. G~. 6 (1965), 212–220. MR322464.
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Baumslag, G. (1974). Finitely Presented Metabelian Groups. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_5
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DOI: https://doi.org/10.1007/978-3-662-21571-5_5
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