Geodesic
Transfinite Lower Central Series of Groups: Parafree Properties and Topological Applications
Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 251-267.

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2-Generated groups $G(i)$, $i=1,2,\dots$, such that $\gamma _{\omega}G(i)\neq\gamma _{\omega+1}G(i)$ are constructed. Moreover, the natural homomorphism $F_2\to G(i)$ of a free group of rank 2 induces isomorphisms $F_2/\gamma _kF_2\simeq G(i)/\gamma _k G(i)$ for all $k\leq q^{i-1}$, where $q$ is a certain prime, and the group $G(1)$ is finitely presented. Methods for realizing a generalized torsion by means of fundamental groups of complements to links are also considered. Torsion-free fundamental groups of 3-manifolds for which the lower central series do not stabilize at the $\omega$th step are constructed. For an arbitrary finite 2-torsion-free abelian group $A$, a 3-manifold (with boundary) is constructed such that $\gamma _{\omega }/\gamma _{\omega +1}\simeq A$ for its fundamental group. 
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R. V. Mikhailov. Transfinite Lower Central Series of Groups: Parafree Properties and Topological Applications. Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 251-267. http://geodesic.mathdoc.fr/item/TRSPY_2002_239_a15/

[1] Baumslag G., “Groups with the same lower central sequences as a relatively free group. I: The groups”, Trans. Amer. Math. Soc., 129 (1967), 308–321 | DOI | MR | Zbl

[2] Baumslag G., “Wreath products and finitely presented groups”, Math. Ztschr., 75 (1961), 22–28 | DOI | MR

[3] Baumslag G., “Groups with the same lower central sequences as a relatively free group. II: Properties”, Trans. Amer. Math. Soc., 142 (1969), 507–538 | DOI | MR | Zbl

[4] Braun K. S., Kogomologii grupp, Nauka, M., 1987 | MR

[5] Cannon J. W., “The recognition problem: What is a topological manifold?”, Bull. Amer. Math. Soc., 84 (1978), 832–866 | DOI | MR | Zbl

[6] Cochran T., Derivatives of links: Milnor's concordance invariants and Massey's products, Mem. Amer. Math. Soc., 84, Amer. Math. Soc., Providence, RI, 1990, 427 | MR

[7] Cochran T., Orr K., “Stability of lower central series of compact 3-manifold groups”, Topology, 37:3 (1998), 497–526 | DOI | MR | Zbl

[8] Dwyer W. G., “Homology, Massey products and maps between groups”, J. Pure and Appl. Algebra, 6 (1975), 177–190 | DOI | MR | Zbl

[9] Freedman M. H., Teichner P., “4-Manifold topology. II: Dwyer's filtration and surgery kernels”, Invent. Math., 122:3 (1995), 531–558 | DOI | MR

[10] Gruenberg K. W., “The residual nilpotence of certain presentations of finite groups”, Arch. Math., 13 (1962), 408–417 | DOI | MR | Zbl

[11] Gutierrez M. A., “Homology and completion of groups”, J. Algebra, 51 (1978), 354–366 | DOI | MR | Zbl

[12] Hartley B., “Topics in the theory of nilpotent groups”, Group theory. Essays for Philip Hall, Acad. Press, London, New York, 1984, 61–120 | MR

[13] Kirby R. C., “Problems in low-dimensional topology”, Geometric topology, Pt. 2, AMS/IP Stud. Adv. Math., 2, ed. W. H. Kazez, Amer. Math. Soc. Intern. Press, Providence, RI, Cambridge, MA, 1997, 35–473 ; http://www.math.berkeley.edu/~kirby/problems.ps.gz | MR | Zbl

[14] Kuzmin Yu. V., “Gomologii grupp vida $F/N'$”, DAN SSSR, 296:2 (1987), 267–270 | MR

[15] Levine J., “Finitely-presented groups with long lower central series”, Israel J. Math., 73 (1991), 57–64 | DOI | MR | Zbl

[16] Levine J., “An approach to homotopy classification of links”, Trans. Amer. Math. Soc., 306:1 (1988), 361–387 | DOI | MR | Zbl

[17] Levine J., “The $\bar\mu$-invariants of based links”, Lect. Notes Math., 1350, 1988, 87–103 | MR | Zbl

[18] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[19] Likhtman A. I., “O nilpotentno-approksimiruemykh gruppakh”, Sib. mat. zhurn., 6:4 (1965), 862–866 | Zbl

[20] Maltsev A. I., “Obobschennye nilpotentnye algebry i assotsiirovannye s nimi gruppy”, Mat. sb., 25 (1949), 347–366

[21] Melikhov S. A., Mikhailov R. V., Finer analogues of Milnor's link groups, arXiv: /abs/math.GT/0201022

[22] Milnor J., “Link groups”, Ann. Math., 59 (1954), 177–195 | DOI | MR | Zbl

[23] Orr K., “Homotopy invariants of links”, Invent. Math., 95 (1989), 379–394 | DOI | MR | Zbl

[24] Papakiryakopulos S. D., “O lemme Dena i asferichnosti uzlov”, Matematika: Sb. per., 2:4 (1958), 23–47

[25] Passi I. B. S., Stambach U., “A filtration of Schur multiplicator”, Math. Ztschr., 135:2 (1974), 143–148 | DOI | MR | Zbl

[26] Shtanko M. A., “Approksimatsiya vlozhenii kompaktov v korazmernosti bolshe dvukh”, DAN SSSR, 198:4 (1971), 783–786

[27] Stallings J., “Homology and central series of groups”, J. Algebra, 2:2 (1965), 170–181 | DOI | MR | Zbl

[28] Stöhr R., “On torsion in free central extensions of some torsion free groups”, J. Pure and Appl. Algebra, 46 (1987), 249–289 | DOI | MR | Zbl