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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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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:
/ 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

## Abstract

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

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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,

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