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Decomposing locally compact groups into simple pieces

Published online by Cambridge University Press:  02 September 2010

PIERRE-EMMANUEL CAPRACE  and
NICOLAS MONOD
PIERRE-EMMANUEL CAPRACE
Affiliation:
UCLouvain, 1348 Louvain-la-Neuve, Belgium. e-mail: [email protected]
NICOLAS MONOD
Affiliation:
EPFL, 1015 Lausanne, Switzerland. e-mail: [email protected]
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Abstract

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple.

Two appendices introduce results and examples around the concept of quasi-product.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 1 , January 2011 , pp. 97 - 128
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[Are46]Arens, R.Topologies for homeomorphism groups. Amer. J. Math. 68 (1946), 593–610.CrossRefGoogle Scholar
[BEW08]Barnea, Y., Ershov, M. and Weigel, T. Abstract commensurators of profinite groups. Trans. AMS (to appear).Google Scholar
[Bir34]Birkhoff, G.The topology of transformation-sets. Ann. of Math. (2) 35 (1934), no. 4, 861–875.CrossRefGoogle Scholar
[BM00]Burger, M. and Mozes, S. Groups acting on trees: from local to global structure. Inst. Hautes Études Sci. Publ. Math. (2000), no. 92, 113–150 (2001).Google Scholar
[BM02]Burger, M. and Monod, N.Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12 (2002), no. 2, 219–280.CrossRefGoogle Scholar
[Bou60]Bourbaki, N.Éléments de mathématique. Première partie. (Fascicule III.) Livre III; Topologie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Troisième édition revue et augmentée, Actualités Sci. Indust., No. 1143. (Hermann, 1960).Google Scholar
[Bou71]Bourbaki, N.Éléments de mathématique. Topologie générale. Chapitres 1 à 4 (Hermann, 1971).Google Scholar
[Bou74]Bourbaki, N.Éléments de mathématique. Topologie générale, chap. 5 à 10 (nouvelle édition) (Hermann, 1974).Google Scholar
[Bra48]Braconnier, J.Sur les groupes topologiques localement compacts. J. Math. Pures Appl. (9) 27 (1948), 1–85.Google Scholar
[Cap09]Caprace, P.-E.Amenable groups and Hadamard spaces with a totally disconnected isometry group. Comment. Math. Helv. 84 (2009), 437–455.CrossRefGoogle Scholar
[CER08]Carbone, L., Ershov, M. and Ritter, G.Abstract simplicity of complete Kac-Moody groups over finite fields. J. Pure Appl. Algebra 212 (2008), no. 10, 2147–2162.CrossRefGoogle Scholar
[CH06]Caprace, P.-E. and Haglund, F.On geometric flats in the CAT(0) realization of Coxeter groups and Tits buildings. Canad. J. Math. 61 (2009), no. 4, 740–761.CrossRefGoogle Scholar
[CM09]Caprace, P.-E. and Monod, N.Isometry groups of non-positively curved spaces: structure theory. J Topology 2 (2009), no. 4, 661–700.CrossRefGoogle Scholar
[DdSMS99]Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D.Analytic Pro-p Groups, second ed. Cambridge Studies in Advanced Math., vol. 61 (Cambridge University Press, 1999). MR MR1720368 (2000m:20039)CrossRefGoogle Scholar
[Dix96]Dixmier, J.Les Algèbres d'Opérateurs dans L'espace Hilbertien (Algèbres de von Neumann). Les Grands Classiques Gauthier-Villars (Éditions Jacques Gabay, 1996), Reprint of the second (1969) edition.Google Scholar
[Dug96]Dugundji, J.Topology (Allyn and Bacon Inc., 1966).Google Scholar
[Fre36]Freudenthal, H.Topologische Gruppen mit genügend vielen fastperiodischen Funktionen. Ann. of Math. (2) 37 (1936), no. 1, 57–77.CrossRefGoogle Scholar
[Gb46]Gol′ berg, P.The Silov II-groups of locally normal groups. Rec. Math. [Mat. Sbornik] N.S. 19 (61) (1946), 451–460.Google Scholar
[GLS94]Gorenstein, D., Lyons, R. and Solomon, R. The classification of the finite simple groups, Math. Surv. Monogr. vol. 40 (American Mathematical Society, 1994).Google Scholar
[GM67]Grosser, S. and Moskowitz, M.On central topological groups. Trans. Amer. Math. Soc. 127 (1967), 317–340.CrossRefGoogle Scholar
[GM71]Grosser, S. and Moskowitz, M.Compactness conditions in topological group. J. Reine Angew. Math. 246 (1971), 1–40.Google Scholar
[GP57]Gleason, A. M. and Palais, R. S.On a class of transformation groups. Amer. J. Math. 79 (1957), 631–648.CrossRefGoogle Scholar
[Gui73]Guivarc'h, Y.Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 333–379.CrossRefGoogle Scholar
[HR79]Hewitt, E. and Ross, K. A.Abstract harmonic analysis. Vol. I, second ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115 (Springer-Verlag, 1979). Structure of topological groups, integration theory, group representations.CrossRefGoogle Scholar
[Iwa49]Iwasawa, K.On some types of topological groups. Ann. of Math. (2) 50 (1949), 507–558.CrossRefGoogle Scholar
[Kel75]Kelley, J. L.General Topology (Springer-Verlag, 1975). Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Math., No. 27.Google Scholar
[KK44]Kakutani, S. and Kodaira, K.Über das Haarsche Mass in der lokal bikompakten Gruppe. Proc. Imp. Acad. Tokyo 20 (1944), 444–450.Google Scholar
[LW70]Lee, D. H. and Wu, T.-S.On CA topological groups. Duke Math. J. 37 (1970), 515–521.CrossRefGoogle Scholar
[Mar91]Margulis, G. A. Discrete subgroups of semisimple Lie groups. Ergeb. Math. Grenzgeb (3), vol. 17 (1991).CrossRefGoogle Scholar
[Men65]Mennicke, J. L.Finite factor groups of the unimodular group. Ann. of Math. (2) 81 (1965), 31–37.CrossRefGoogle Scholar
[Mon01]Monod, N.Continuous bounded cohomology of locally compact groups. Lecture Notes in Math. vol. 1758 (Springer-Verlag, 2001).CrossRefGoogle Scholar
[Moz98]Mozes, S.Products of trees, lattices and simple groups. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), no. Extra Vol. II (1998), pp. 571–582 (electronic). MR MR1648106 (2000a:20056).Google Scholar
[MŚ59]Macbeath, M. and Świerczkowski, S.On the set of generators of a subgroup. Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 280–281.CrossRefGoogle Scholar
[MvN43]Murray, F. J. and Neumann, J. vonOn rings of operators, IV. Ann. of Math. (2) 44 (1943), 716–808.CrossRefGoogle Scholar
[MZ55]Montgomery, D. and Zippin, L. Topological Transformation Groups (Interscience Publishers, 1955).Google Scholar
[Nik09]Nikolov, N. Strange images of profinite groups. Preprint, 2009.Google Scholar
[Pat88]Paterson, A. L. T. Amenability. Math. Surv. Monogr., vol. 29 (1988).CrossRefGoogle Scholar
[Pla65]Platonov, V. P.Lokal projective nilpotent radicals in topological groups. Dokl. Akad. Nauk BSSR 9 (1965), 573–577.Google Scholar
[Pla77]Plaumann, P.Automorphismengruppen diskreter Gruppen als topologische Gruppen. Arch. Math. (Basel) 29 (1977), no. 1, 32–33.CrossRefGoogle Scholar
[Pon39]Pontrjagin, L. S.Topological Groups. Princeton Mathematical Series, v. 2 (Princeton University Press, 1939). Translated from the Russian by Emma Lehmer.Google Scholar
[Sch25]Schreier, O.Abstrakte kontinuierliche Gruppen. Abh. Math. Hamburg 4 (1925), 15–32.CrossRefGoogle Scholar
[Tit64]Tits, J.Algebraic and abstract simple groups. Ann. of Math. (2) 80 (1964), 313–329.CrossRefGoogle Scholar
[TV99]Thomas, S. and Velickovic, B.On the complexity of the isomorphism relation for finitely generated groups. J. Algebra 217 (1999), no. 1, 352–373.CrossRefGoogle Scholar
[Uša63]Ušakov, V. I.Topological FC-groups. Sibirsk. Mat. Ž. 4 (1963), 1162–1174.Google Scholar
[vD30]van Dantzig, D.Ueber topologisch homogene Kontinua. Fund. Math. 15 (1930), 102–125.CrossRefGoogle Scholar
[vD31]van Dantzig, D. Studien over topologische algebra (proefschrift). Ph.D. thesis (Groningen, 1931).Google Scholar
[vDvdW28]van Dantzig, D. and van der Waerden, B. L.über metrisch homogene Räume. Abh. Math. Hamburg 6 (1928), no. 2, 367–376.CrossRefGoogle Scholar
[Vie27]Vietoris, L.Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann. 97 (1927), 454–472.CrossRefGoogle Scholar
[Wan69]Wang, S. P.The automorphism group of a locally compact group. Duke Math. J. 36 (1969), 277–282.CrossRefGoogle Scholar
[Wei40]Weil, A. L'intégration dans les groupes topologiques et ses applications. Actual. Sci. Ind. no. 869 (Hermann et Cie., 1940).Google Scholar
[Wil71]Wilson, J. S.Groups with every proper quotient finite. Proc. Camb. Philos. Soc. 69 (1971), 373–391.CrossRefGoogle Scholar
[Wil94]Willis, G. A.The structure of totally disconnected, locally compact groups. Math. Ann. 300 (1994), no. 2, 341–363.CrossRefGoogle Scholar
[Wil07]Willis, G. A.Compact open subgroups in simple totally disconnected groups. J. Algebra 312 (2007), no. 1, 405–417.CrossRefGoogle Scholar
[Wu71]Wu, T.-S.On (CA) topological groups. II. Duke Math. J. 38 (1971), 513–519.CrossRefGoogle Scholar
[Zer76]Zerling, D.(CA) topological groups. Proc. Amer. Math. Soc. 54 (1976), 345–351.Google Scholar