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Forum Mathematicum

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Volume 31, Issue 2


Dimension quotients, Fox subgroups and limits of functors

Roman Mikhailov
  • Corresponding author
  • Steklov Mathematical Institute, Laboratory of Modern Algebra and Applications, St. Petersburg State University, 14th Line, 29b, Saint Petersburg 199178, Russia; and School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Inder Bir S. Passi
  • Centre for Advanced Study in Mathematics, Panjab University, Sector 14, Chandigarh 160014; and Indian Institute of Science Education and Research, Mohali (Punjab) 140306 India
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Published Online: 2018-10-16 | DOI: https://doi.org/10.1515/forum-2017-0226


This paper presents a description of the fourth dimension quotient, using the theory of limits of functors from the category of free presentations of a given group to the category of abelian groups. A functorial description of a quotient of the third Fox subgroup is given and, as a consequence, an identification (not involving an isolator) of the third Fox subgroup is obtained. It is shown that the limit over the category of free representations of the third Fox quotient represents the composite of two derived quadratic functors.

Keywords: Group ring; dimension subgroup; derived functor

MSC 2010: 20C05; 20C07; 18A22; 18G10; 18G50; 16E40


  • [1]

    J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud. 82, Princeton University Press, Princeton, 1974. Google Scholar

  • [2]

    L. Breen, On the functorial homology of abelian groups, J. Pure Appl. Algebra 142 (1999), no. 3, 199–237. CrossrefGoogle Scholar

  • [3]

    P. M. Cohn, Generalization of a theorem of Magnus, Proc. Lond. Math. Soc. (3) 2 (1952), 297–310. Google Scholar

  • [4]

    A. Dold and D. Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier Grenoble 11 (1961), 201–312. CrossrefGoogle Scholar

  • [5]

    I. Emmanouil and R. Mikhailov, A limit approach to group homology, J. Algebra 319 (2008), no. 4, 1450–1461. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    R. H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560. CrossrefGoogle Scholar

  • [7]

    K. W. Gruenberg, Cohomological Topics in Group Theory, Lecture Notes in Math. 143, Springer, Berlin, 1970. Google Scholar

  • [8]

    N. Gupta, Free Group Rings, Contemp. Math. 66, American Mathematical Society, Providence, 1987. Google Scholar

  • [9]

    M. Hartl, R. Mikhailov and I. B. S. Passi, Dimension quotients, J. Indian Math. Soc. (N.S.) Centenary Vol. (2007), 63–107. Google Scholar

  • [10]

    S. O. Ivanov and R. Mikhailov, A higher limit approach to homology theories, J. Pure Appl. Algebra 219 (2015), no. 6, 1915–1939. CrossrefWeb of ScienceGoogle Scholar

  • [11]

    F. Jean, Foncteurs derives de lalgebre symetrique: Application au calcul de certains groupes dhomologie fonctorielle des espaces K(B, n), Doctoral thesis, University of Paris 13, 2002, http://homepages.abdn.ac.uk/mth192/pages/html/archive/jean.html.

  • [12]

    R. Karan, D. Kumar and L. R. Vermani, Some intersection theorems and subgroups determined by certain ideals in integral group rings. II, Algebra Colloq. 9 (2002), no. 2, 135–142. Google Scholar

  • [13]

    B. Köck, Computing the homology of Koszul complexes, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3115–3147. CrossrefGoogle Scholar

  • [14]

    G. Losey, On dimension subgroups, Trans. Amer. Math. Soc. 97 (1960), 474–486. CrossrefGoogle Scholar

  • [15]

    S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Grad. Texts in Math. 5, Springer, New York, 1998. Google Scholar

  • [16]

    W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. 111 (1935), no. 1, 259–280. CrossrefGoogle Scholar

  • [17]

    W. Magnus, Über Beziehungen zwischen höheren Kommutatoren, J. Reine Angew. Math. 177 (1937), 105–115. Google Scholar

  • [18]

    R. Mikhailov and I. B. S. Passi, Lower Central and Dimension Series of Groups, Lecture Notes in Math. 1952, Springer, Berlin, 2009. Google Scholar

  • [19]

    R. Mikhailov and I. B. S. Passi, Free group rings and derived functors, Proceedings of the 7th European Congress of Mathematics, EMS Publishing House, Zurich (2018), 407–425. Google Scholar

  • [20]

    R. Mikhailov and I. B. S. Passi, Generalized dimension subgroups and derived functors, J. Pure Appl. Algebra 220 (2016), no. 6, 2143–2163. CrossrefWeb of ScienceGoogle Scholar

  • [21]

    I. B. S. Passi, Dimension subgroups, J. Algebra 9 (1968), 152–182. Web of ScienceCrossrefGoogle Scholar

  • [22]

    I. B. S. Passi, Group Rings and Their Augmentation Ideals, Lecture Notes in Math. 715, Springer, Berlin, 1979. Google Scholar

  • [23]

    D. Quillen, Cyclic cohomology and algebra extensions, K-Theory 3 (1989), no. 3, 205–246. CrossrefGoogle Scholar

  • [24]

    E. Rips, On the fourth integer dimension subgroup, Israel J. Math. 12 (1972), 342–346. CrossrefGoogle Scholar

  • [25]

    K. I. Tahara, On the structure of Q3(G) and the fourth dimension subgroups, Japan. J. Math. (N.S.) 3 (1977), no. 2, 381–394. Google Scholar

  • [26]

    K.-I. Tahara, The fourth dimension subgroups and polynomial maps, J. Algebra 45 (1977), no. 1, 102–131. CrossrefGoogle Scholar

  • [27]

    E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937), 152–160. Google Scholar

  • [28]

    I. A. Yunus, A problem of Fox, Soviet Math. Dokl. 30 (1984), 346–350. Google Scholar

About the article

Received: 2017-10-25

Revised: 2018-09-13

Published Online: 2018-10-16

Published in Print: 2019-03-01

Funding Source: Russian Science Foundation

Award identifier / Grant number: 16-11-10073

The research is supported by the Russian Science Foundation grant N 16-11-10073.

Citation Information: Forum Mathematicum, Volume 31, Issue 2, Pages 385–401, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0226.

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