Algebraic & Geometric Topology

Torsion homology and cellular approximation

Ramón Flores and Fernando Muro

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Abstract

We describe the role of the Schur multiplier in the structure of the p –torsion of discrete groups. More concretely, we show how the knowledge of H 2 G allows us to approximate many groups by colimits of copies of p –groups. Our examples include interesting families of noncommutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of Emmanuel Farjoun.

Article information

Source
Algebr. Geom. Topol., Volume 19, Number 1 (2019), 457-476.

Dates
Received: 27 March 2018
Revised: 3 September 2018
Accepted: 11 September 2018
First available in Project Euclid: 12 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.agt/1549940438

Digital Object Identifier
doi:10.2140/agt.2019.19.457

Mathematical Reviews number (MathSciNet)
MR3910586

Zentralblatt MATH identifier
07053579

Subjects
Primary: 20F99: None of the above, but in this section 55P60: Localization and completion

Keywords
Torsion homology cellular group

Citation

Flores, Ramón; Muro, Fernando. Torsion homology and cellular approximation. Algebr. Geom. Topol. 19 (2019), no. 1, 457--476. doi:10.2140/agt.2019.19.457. https://projecteuclid.org/euclid.agt/1549940438


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References

  • S I Adian, V S Atabekyan, Central extensions of free periodic groups, Mat. Sb. 209 (2018) 3–16 In Russian; translation to appear in Sb. Math. 209
  • I S Ashmanov, A Y Olshanskiĭ, Abelian and central extensions of aspherical groups, Izv. Vyssh. Uchebn. Zaved. Mat. (1985) 48–60 In Russian; translated in Soviet Math. 29 (1985) 65–82
  • L Bartholdi, Endomorphic presentations of branch groups, J. Algebra 268 (2003) 419–443
  • A J Berrick, M M Matthey, Homological realization of prescribed abelian groups via $K$–theory, Math. Proc. Cambridge Philos. Soc. 142 (2007) 249–258
  • M Blomgren, W Chachólski, E D Farjoun, Y Segev, Idempotent deformations of finite groups, preprint (2012)
  • A K Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133–150
  • A K Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994) 831–873
  • A K Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997) 1321–1354
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)
  • J Buckner, M Dugas, Co-local subgroups of abelian groups, from “Abelian groups, rings, modules, and homological algebra” (P Goeters, O M G Jenda, editors), Lect. Notes Pure Appl. Math. 249, Chapman & Hall, Boca Raton, FL (2006) 29–37
  • C Casacuberta, On structures preserved by idempotent transformations of groups and homotopy types, from “Crystallographic groups and their generalizations” (P Igodt, H Abels, Y Félix, F Grunewald, editors), Contemp. Math. 262, Amer. Math. Soc., Providence, RI (2000) 39–68
  • N Castellana, J A Crespo, J Scherer, Deconstructing Hopf spaces, Invent. Math. 167 (2007) 1–18
  • W Chachólski, On the functors $\mathrm{CW}_A$ and $P_A$, Duke Math. J. 84 (1996) 599–631
  • W Chachólski, E Damian, E D Farjoun, Y Segev, The $A$–core and $A$–cover of a group, J. Algebra 321 (2009) 631–666
  • W Chachólski, E D Farjoun, R Flores, J Scherer, Cellular properties of nilpotent spaces, Geom. Topol. 19 (2015) 2741–2766
  • I Chatterji (editor), Guido's book of conjectures, Monographies de L'Enseignement Mathématique 40, Enseign. Math., Geneva (2008)
  • C Chou, Elementary amenable groups, Illinois J. Math. 24 (1980) 396–407
  • M M Day, Amenable semigroups, Illinois J. Math. 1 (1957) 509–544
  • W G Dwyer, The centralizer decomposition of $BG$, from “Algebraic topology: new trends in localization and periodicity” (C Broto, C Casacuberta, G Mislin, editors), Progr. Math. 136, Birkhäuser, Basel (1996) 167–184
  • W G Dwyer, J P C Greenlees, S Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357–402
  • E D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer (1996)
  • E D Farjoun, R Göbel, Y Segev, Cellular covers of groups, J. Pure Appl. Algebra 208 (2007) 61–76
  • E D Farjoun, R Göbel, Y Segev, S Shelah, On kernels of cellular covers, Groups Geom. Dyn. 1 (2007) 409–419
  • R J Flores, Nullification and cellularization of classifying spaces of finite groups, Trans. Amer. Math. Soc. 359 (2007) 1791–1816
  • R J Flores, R M Foote, The cellular structure of the classifying spaces of finite groups, Israel J. Math. 184 (2011) 129–156
  • R Flores, J Scherer, Cellular covers of local groups, Mediterr. J. Math. 15 (2018) 15:229
  • L Fuchs, Cellular covers of totally ordered abelian groups, Math. Slovaca 61 (2011) 429–438
  • L Fuchs, R Göbel, Cellular covers of abelian groups, Results Math. 53 (2009) 59–76
  • R Göbel, Cellular covers for $R$–modules and varieties of groups, Forum Math. 24 (2012) 317–337
  • R I Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984) 939–985 In Russian; translated in Math. USSR-Izv. 25 (1985) 259–300
  • R I Grigorchuk, On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata, from “Groups St Andrews 1997 in Bath, I” (C M Campbell, E F Robertson, N Ruskuc, G C Smith, editors), London Math. Soc. Lecture Note Ser. 260, Cambridge Univ. Press (1999) 290–317
  • N Gupta, S Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983) 385–388
  • G Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Series 2, Clarendon, New York (1987)
  • S ÓhÓgáin, On torsion-free minimal abelian groups, Comm. Algebra 33 (2005) 2339–2350
  • A Y Olshanskiĭ, Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series) 70, Kluwer, Dordrecht (1991)
  • M Petapirak, Cellular covers and varieties of groups, Dissertation, Universität Duisburg-Essen (2014) \setbox0\makeatletter\@url https://core.ac.uk/download/pdf/33797402.pdf {\unhbox0
  • D C Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128, Princeton Univ. Press (1992)
  • D J S Robinson, A course in the theory of groups, 2nd edition, Graduate Texts in Mathematics 80, Springer (1996)
  • J L Rodríguez, J Scherer, Cellular approximations using Moore spaces, from “Cohomological methods in homotopy theory” (J Aguadé, C Broto, C Casacuberta, editors), Progr. Math. 196, Birkhäuser, Basel (2001) 357–374
  • J L Rodríguez, J Scherer, A connection between cellularization for groups and spaces via two-complexes, J. Pure Appl. Algebra 212 (2008) 1664–1673
  • J L Rodríguez, L Strüngmann, On cellular covers with free kernels, Mediterr. J. Math. 9 (2012) 295–304
  • J L Rodríguez, L Strüngmann, Cellular covers of $\aleph_1$–free abelian groups, J. Algebra Appl. 14 (2015) art. id. 1550139
  • L Strüngmann, Minimal completely decomposable groups, Math. Proc. R. Ir. Acad. 105A (2005) 107–110