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The true prosoluble completion of a group: examples and open problems. (English) Zbl 1138.20027

This carefully structured report begins with a general introduction to topological completions of groups with respect to directed sets of normal subgroups. In particular, this leads to the notion of the ‘true prosoluble completion’ of a group \(\Gamma\) which can be realised as the inverse limit of the projective system of all – not necessarily finite – soluble quotients of \(\Gamma\). (For instance, the true prosoluble completion of the infinite cyclic group \(\mathbb{Z}\) is again \(\mathbb{Z}\) equipped with the discrete topology, whereas the more frequently encountered prosoluble completion of \(\mathbb{Z}\), based upon all finite soluble images of \(\mathbb{Z}\), coincides with the familiar profinite completion \(\widehat\mathbb{Z}\).) Similarly, the ‘true pronilpotent completion’ of a group is the inverse limit of the projective system of all nilpotent quotients of the group, and should not be mixed up with the more frequently encountered pronilpotent completion. The authors provide several examples, and they state various observations and open questions surrounding the profinite, pro-\(p\), prosoluble, true prosoluble and true pronilpotent completions of groups.
The main focus of the report is on analogues of the following fundamental problem suggested by work of A. Grothendieck [Manuscr. Math. 2, 375-396 (1970; Zbl 0239.20065)] on linear representations and profinite completions of discrete groups: given a homomorphism \(\psi\colon\Gamma\to\Delta\) between residually finite groups which induces an isomorphism \(\widehat\psi\colon\widehat\Gamma\to\widehat\Delta\) between the respective profinite completions, what extra information is required to deduce that \(\psi\) itself is an isomorphism between the original groups? Addressing Grothendieck’s more precise original questions, M. R. Bridson and F. J. Grunewald [Ann. Math. (2) 160, No. 1, 359-373 (2004; Zbl 1083.20023)] have shown that the extra information that \(\Gamma\) and \(\Delta\) are finitely presented is not sufficient.
In the paper under review the authors point out how residually nilpotent groups which are parafree, first discovered by G. Baumslag [Bull. Am. Math. Soc. 73, 621-622 (1967; Zbl 0153.35001)], yield negative examples concerning the analogue of Grothendieck’s problem for true pronilpotent completions. They also explain a possible approach to produce negative examples concerning the analogue of Grothendieck’s problem for true prosoluble completions. This approach is based on the (as yet unproven) hypothesis that there exists a finitely generated subgroup \(\Delta\) of the pro-\(2\) completion of the first Grigorchuk group \(\mathcal G\) which contains \(\mathcal G\) properly and has the ‘congruence extension property’. In an appendix one finds a brief description of the first Grigorchuk group and an indication of how candidates for such a group \(\Delta\) can be constructed.
The report is divided into seven sections and one appendix. The individual section headings are: 1. Introduction, 2. Completion with respect to a directed set of normal subgroups, 3. Universal property, 4. Examples of directed sets of normal subgroups, 5. True prosoluble completions, 6. Examples, 7. On the true prosoluble and the true pronilpotent analogues of Grothendieck’s problem, 8. Appendix: Construction of elements in the closure of Grigorchuk group. The report ends with a substantial list of references.

MSC:

20E18 Limits, profinite groups
20E26 Residual properties and generalizations; residually finite groups
20E08 Groups acting on trees
20F14 Derived series, central series, and generalizations for groups
20F19 Generalizations of solvable and nilpotent groups
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[1] Abels H. (1979) An example of a finitely presented solvable group. In: Wall C.T.C. (ed) Homological Group Theory. Durham 1977, Cambridge Univ. Press, pp. 205–211
[2] Barnea Y., Shalev A. (1997) Hausdorff dimension, pro-p groups, and Kac-Moody algebras. Trans. Amer. Math. Soc. 349(12): 5073–5091 · Zbl 0892.20020 · doi:10.1090/S0002-9947-97-01918-1
[3] Baumslag G. (1967) Some groups that are just free. Bull. Amer. Math. Soc. 73, 621–622 · Zbl 0153.35001 · doi:10.1090/S0002-9904-1967-11800-7
[4] Baumslag G. (1968) More groups that just about free. Bull. Amer. Math. Soc. 74, 752–754 · Zbl 0176.29702 · doi:10.1090/S0002-9904-1968-12029-4
[5] Baumslag G. (1969) A non-cyclic one-relator group all of whose finite quotients are cyclic. J. Austral. Math. Soc. 10, 497–498 · Zbl 0214.27402 · doi:10.1017/S1446788700007783
[6] Baumslag G. (1971) Positive one–relator groups. Trans. Amer. Math. Soc. 156, 165–183 · Zbl 0256.20043 · doi:10.1090/S0002-9947-1971-0274562-8
[7] Baumslag, G.: Parafree groups. Bartholdi, L., Ceccherini-Silberstein, T., Smirnova-Nagnibeda, T., Zuk, A. (eds.) Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, pp. 1–15. Series: Progress in Mathematics, vol. 248. Birkhäuser (2005) · Zbl 1083.20500
[8] Birkhoff G. (1937) More–Smith convergence in general topology. Ann. Math. 38, 39–56 · Zbl 0016.08502 · doi:10.2307/1968508
[9] Bourbaki N.: Topologie générale, chapitres 3 et 4, troisième édition. Hermann (1960)
[10] Bourbaki N. (1970) Algèbre, chapitres 1 à 3. Diffusion C.C.L.S., Paris · Zbl 0211.02401
[11] Bourbaki, N.: Groupes et algèbres de Lie, chapitres 2 et 3. Hermann (1972) · Zbl 0244.22007
[12] Bourbaki, N.: Groupes et algèbres de Lie, chapitre 9. Masson (1982) · Zbl 0505.22006
[13] Bridson M., Grunewald F.J. (2004) Grothendieck’s problems concerning profinite completions and representations of groups. Ann. Math. 160, 359–373 · Zbl 1083.20023 · doi:10.4007/annals.2004.160.359
[14] Brunner A.M., Burns R.G., Wiegold J. (1979) On the number of quotients, of one way or another,of the modular group. Math. Sci. 4, 93–98 · Zbl 0409.20030
[15] Brunner A.M., Sidki S., Vieira A.C. (1999) A just nonsolvable torsion-free group defined on the binary tree. J. Algebra 211, 99–114 · Zbl 0920.20029 · doi:10.1006/jabr.1998.7579
[16] Clair B. (1999) Residual amenability and the approximation of L 2-invariants. Mich. Math. J. 46, 331–346 · Zbl 0967.58017 · doi:10.1307/mmj/1030132414
[17] Cochran, T., Harvey, S.: Homology and derived series of groups. arXiv:math.GT/0407203 · Zbl 1159.57004
[18] Dixon, J.D., du Sautoy, M., Mann, A., Segal, D.: Analytic Pro-p Groups. Cambridge University Press (1991) · Zbl 0744.20002
[19] Elek G., Szabo, E.: On sofic groups. arXiv:math.GR/0305352
[20] Glasner, Y., Souto, J., Storm, P.: Finitely generated subgroups of lattices in PSL 2 C. arXiv:math.GT/0504441 · Zbl 1220.20018
[21] Grigorchuk R.I. (1980) Burnside’s problem on periodic groups. Funct. Anal. Appl. 14, 41–43 · Zbl 0595.20029
[22] Grigorchuk, R.I.: Just infinite branch groups. In: du Sautoy, M., Segal, D., Shalev, A. (eds.) New Horizons in Pro-p-groups, pp. 121–179. Brikhäuser (2000) · Zbl 0982.20024
[23] Grigorchuk, R.I.: Solved and unsolved problems around one group. Prog. Math. 117–217 (2005) · Zbl 1165.20021
[24] Grigorchuk R.I., Nekrashevich V.V., Sushchanskiĭ V.I. (2000) Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231, 134–214 · Zbl 1155.37311
[25] Grothendieck A. (1970) Représentations linéaires et compactification profinie des groupes discrets. Manuscripta Math. 2, 375–396 · Zbl 0239.20065 · doi:10.1007/BF01719593
[26] Gruenberg L. (1957) Residual properties of infinite soluble groups. Proc. London Math. Soc. 7, 29–62 · Zbl 0077.02901 · doi:10.1112/plms/s3-7.1.29
[27] Hall M. (1950) A topology for free groups and related groups. Ann. Math. 52, 127–139 · Zbl 0045.31204 · doi:10.2307/1969513
[28] Hall P. (1959) On the finiteness of certain soluble groups. Proc. London Math. Soc. 9, 595–622 [= Collected Works, 515–544] · Zbl 0091.02501 · doi:10.1112/plms/s3-9.4.595
[29] Hall P. (1961) The Frattini subgroups of finitely generated groups. Proc. London Math. Soc. 11, 327–352 [= Collected Works, 581–608] · Zbl 0104.02201 · doi:10.1112/plms/s3-11.1.327
[30] de la Harpe, P.: Topics in Geometric Group Theory. The University of Chicago Press (2000) · Zbl 0965.20025
[31] Higman G., Stone A.H. (1954) On inverse systems with trivial limits. J. London Math. Soc. 29, 233–236 · Zbl 0055.02503 · doi:10.1112/jlms/s1-29.2.233
[32] Kahrobaei, D.: On the residual solvability of generalized free products of finitely generated nilpotent groups. arXiv:math.GR/0510465 · Zbl 1222.20026
[33] Kahrobaei, D.: Are doubles of residually solvable groups. residually solvable? Preprint · Zbl 1183.20027
[34] Kassabov, M., Nikolov, N.: Cartesian products as profinite completions. arXiv:math.GR/ 0602446 · Zbl 1129.20019
[35] Kelley, J.L.: General Topology. Van Nostrand (1955) · Zbl 0066.16604
[36] MacDuffee, C.C.: The Theory of Matrices. Chelsea Publ. Comp. (1946) · Zbl 0007.19507
[37] Magnus, W., Karras, A., Solitar, D.: Combinatorial Group Theory. J. Wiley (1966) · Zbl 0138.25604
[38] Mal’cev, A.I.: On the faithful representation of infinite groups by matrices. Amer. Math. Soc. Transl. 45(2), [Russian original: Mat. SS.(N.S.) 8(50) (1940), pp. 405–422]
[39] Mal’cev A.I. (1949) Generalized nilpotent algebras and their associated groups. Mat. Sbornik N.S. 25(67): 347–366
[40] Nikolov N., Segal D. (2003) Finite index subgroups in profinite groups. C.R. Acad. Sci. Paris, Sér. I 337, 303–308 · Zbl 1033.20029
[41] Nikolov, N., Segal, D.: On finitely generated profinite groups I: strong completeness and uniform bounds. arXiv:math.GR/0604399 · Zbl 1126.20018
[42] Nikolov, N., Segal, D.: On finitely generated profinite groups II: products in quasisimple groups. arXiv:math.GR/0604400 · Zbl 1126.20018
[43] Peterson L.H. (1973) Discontinuous characters and subgroups of finite index. Pacific J. Math. 44, 683–691 · Zbl 0263.22004
[44] Platonov V.P., Tavgen O.I. (1986) On Grothendieck’s problem of profinite completions of groups. Soviet Math. Dokl. 33, 822–825 · Zbl 0614.20016
[45] Platonov V.P., Tavgen O.I. (1990) Grothendieck’s problem on profinite completions and representations of groups. K-theory 4, 89–101 · Zbl 0722.20020 · doi:10.1007/BF00534194
[46] Raptis E., Varsos D. (1989) Residual properties of HNN-extensions with base group an abelian group. J. Pure Appl. Algebra 59, 285–290 · Zbl 0676.20012 · doi:10.1016/0022-4049(89)90098-4
[47] Ribes, L., Zalesskii, P.: Profinite Groups. Springer (2000) · Zbl 0949.20017
[48] Robinson, D.J.S.: A course in the theory of groups. Springer (1982) · Zbl 0483.20001
[49] du Sautoy, M., Segal, D., Shalev, A.: New Horizons in Pro-p-Groups. Birkhäuser (2000) · Zbl 0945.00009
[50] Serre, J.–P. Cohomologie Galoisienne. Lecture Notes in Math. 5, Springer (1973) [cinquième édition 1994]
[51] Sidki S. (2004) Finite automata of polynomial growth do not generate a free group. Geom. Dedicata 108, 193–204 · Zbl 1075.20011 · doi:10.1007/s10711-004-2368-0
[52] Stallings J. (1965) Homology of central series of groups. J. of Algebra 2, 170–181 · Zbl 0135.05201 · doi:10.1016/0021-8693(65)90017-7
[53] Weil, A: L’intégration dans les groupes topologiques et ses applications. Hermann (1940) · Zbl 0063.08195
[54] Wilson, J.: Profinite Groups. Clarendon Press (1998) · Zbl 0909.20001
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