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Infinite presentability of groups and condensation

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Connections with homological algebra and category theory

Published online by Cambridge University Press:  02 January 2014

Robert Bieri ,
Yves Cornulier ,
Luc Guyot  and
Ralph Strebel
Robert Bieri
Affiliation:
Department of Mathematics, Johann Wolfgang Goethe-Universität Frankfurt, 60054 Frankfurt am Main, Germany ([email protected])
Yves Cornulier
Affiliation:
CNRS and Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay, France ([email protected])
Luc Guyot
Affiliation:
STI, EPFL, INN 238 Station 14, Lausanne 1015, Switzerland ([email protected])
Ralph Strebel
Affiliation:
Département de Mathématiques, Chemin du Musée 23, Université de Fribourg, 1700 Fribourg, Switzerland ([email protected])
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Abstract

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor–Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

Keywords

Cantor–Bendixson rankcondensation groupinfinitely presented metabelian groupinvariant SigmaThompson’s group Fspace of marked groups

MSC classification

Secondary: 20F05: Generators, relations, and presentations 20E05: Free nonabelian groups 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F16: Solvable groups, supersolvable groups 20J06: Cohomology of groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 4 , October 2014 , pp. 811 - 848
Copyright
©Cambridge University Press 2013 

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References

Abels, H., An example of a finitely presented solvable group, in Homological Group Theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., Volume 36, pp. 205–211 (Cambridge Univ. Press, Cambridge, 1979).CrossRefGoogle Scholar
Abels, H., Finite presentability of $S$-arithmetic groups, in Compact presentability of solvable groups, Lecture Notes in Mathematics, Volume 1261 (Springer-Verlag, Berlin, 1987).Google Scholar
Abels, H. and Brown, K. S., Finiteness properties of solvable $S$-arithmetic groups: an example, J. Pure Appl. Algebra 44 (1987), 77–83.CrossRefGoogle Scholar
Adjan, S. I., Infinite irreducible systems of group identities, Dokl. Akad. Nauk SSSR 190 (1970), 499–501.Google Scholar
Bass, H., Some remarks on group actions on trees, Comm. Algebra 4 (12) (1976), 1091–1126.Google Scholar
Baumslag, G., Wreath products and finitely presented groups, Math. Z. 75 (1960/1961), 22–28.Google Scholar
Baumslag, G., Gildenhuys, D. and Strebel, R., Algorithmically insoluble problems about finitely presented solvable groups, Lie and associative algebras. I, J. Pure Appl. Algebra 39 (1–2) (1986), 53–94.CrossRefGoogle Scholar
Bieri, R. and Groves, J. R. J., The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347 (1984), 168–195.Google Scholar
Bieri, R., Neumann, W. D. and Strebel, R., A geometric invariant of discrete groups, Invent. Math. 90 (3) (1987), 451–477.Google Scholar
Bridson, M. R. and Miller, C. F. III, Structure and finiteness properties of subdirect products of groups, Proc. Lond. Math. Soc. (3) 98 (3) (2009), 631–651.Google Scholar
Boyer, D. L., Enumeration theorems in infinite Abelian groups, Proc. Amer. Math. Soc. 7 (1956), 565–570.Google Scholar
Bieri, R. and Strebel, R., Almost finitely presented soluble groups, Comment. Math. Helv. 53 (2) (1978), 258–278.CrossRefGoogle Scholar
Bieri, R. and Strebel, R., Valuations and finitely presented metabelian groups, Proc. Lond. Math. Soc. (3) 41 (3) (1980), 439–464.CrossRefGoogle Scholar
Baumslag, G., Strebel, R. and Thomson, M. W., On the multiplicator of $F/ {\gamma }_{c} R$, J. Pure Appl. Algebra 16 (2) (1980), 121–132.Google Scholar
Camm, R., Simple free products, J. Lond. Math. Soc. 1 (1) (1953), 66.Google Scholar
Cannon, J. W., Floyd, W. J. and Parry, W. R., Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (3–4) (1996), 215–256.Google Scholar
Chabauty, C., Limite d’ensembles et géométrie des nombres, Bull. Soc. Math. France 78 (1950), 143–151.Google Scholar
Champetier, C. and Guirardel, V., Limit groups as limits of free groups, Israel J. Math. 146 (2005), 1–75.Google Scholar
Cornulier, Y., Guyot, L. and Pitsch, W., On the isolated points in the space of groups, J. Algebra 307 (1) (2007), 254–277.CrossRefGoogle Scholar
Cornulier, Y., Guyot, L. and Pitsch, W., The space of subgroups of an abelian group, J. Lond. Math. Soc. (2) 81 (3) (2010), 727–746.Google Scholar
Cornulier, Y., Infinite conjugacy classes in groups acting on trees, Groups Geom. Dyn. 3 (2) (2009), 267–277.CrossRefGoogle Scholar
Cornulier, Y., Finitely presented wreath products and double coset decompositions, Geometriae Dedicata 122 (1) (2006), 89–108.Google Scholar
Cornulier, Y., On the Cantor-Bendixson rank of metabelian groups, Ann. Inst. Fourier (Grenoble) 61 (2) (2011), 593–618.CrossRefGoogle Scholar
Cornulier, Y., A sofic group away from amenable groups, Mathematische Annalen 350 (2) (2011), 269–275.Google Scholar
Cornulier, Y., Stalder, Y. and Valette, A., Proper actions of wreath products and generalizations, Trans. Amer. Math. Soc 364 (2012), 3159–3184.Google Scholar
Culler, M. and Morgan, J. W., Group actions on $ \mathbb{R} $-trees, Proc. Lond. Math. Soc. (3) 55 (3) (1987), 571–604.CrossRefGoogle Scholar
Culler, M. and Vogtmann, K., A group-theoretic criterion for property FA, Proc. Amer. Math. Soc. 124 (3) (1996), 677–683.Google Scholar
Delzant, T., Sous-groupes distingués et quotients des groupes hyperboliques, Duke Math. J. 83 (3) (1996), 661–682.CrossRefGoogle Scholar
Greendlinger, M., An analogue of a theorem of Magnus, Arch. Math. 12 (1) (1961), 94–96.Google Scholar
Grigorchuk, R. I., Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 25 (2) (1984), 259–300.Google Scholar
Grigorchuk, R. I., On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata, in Groups St. Andrews 1997 in Bath, I, London Mathematical Society Lecture Note Series, Volume 260, pp. 290–317 (Cambridge Univ. Press, Cambridge, 1999).Google Scholar
Grigorchuk, R., Solved and unsolved problems around one group, in Infinite groups: geometric, combinatorial and dynamical aspects, Progress in Mathematics, Volume 248, pp. 117–218 (Birkhäuser, Basel, 2005).Google Scholar
Hall, P., Finiteness conditions for soluble groups, Proc. Lond. Math. Soc. (3) 4 (1954), 419–436.Google Scholar
Hall, P., The Frattini subgroups of finitely generated groups, Proc. Lond. Math. Soc. (3) 11 (1961), 327–352.Google Scholar
Hilton, P. J. and Stammbach, U., A course in homological algebra, second ed., Graduate Texts in Mathematics, Volume 4 (Springer-Verlag, New York, 1997).CrossRefGoogle Scholar
Humphreys, J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Volume 29 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
Kleĭman, Ju. G., On some questions of the theory of varieties of groups, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1) (1983), 37–74.Google Scholar
Lyndon, R. C. and Schupp, P. E., Combinatorial group theory, in Classics in mathematics. (Springer-Verlag, Berlin, 1977).Google Scholar
Lyulko, A. N., Normal subgroups of Abels groups, Mat. Zametki 36 (3) (1984), 289–294.Google Scholar
Mann, A., A note on recursively presented and co-recursively presented groups, Bull. Lond. Math. Soc. 14 (2) (1982), 112–118.CrossRefGoogle Scholar
McCarthy, D., Infinite groups whose proper quotient groups are finite. I, Comm. Pure Appl. Math. 21 (1968), 545–562.Google Scholar
Mennicke, J. L., Finite factor groups of the unimodular group, Ann. of Math. (2) 81 (1965), 31–37.CrossRefGoogle Scholar
Neumann, B. H., Some remarks on infinite groups, Proc. Lond. Math. Soc. (2) 12 (1937), 120–127.CrossRefGoogle Scholar
Ould Houcine, A., Embeddings in finitely presented groups which preserve the center, J. Algebra 307 (1) (2007), 1–23.CrossRefGoogle Scholar
Ol’šanskiĭ, A. Ju., The finite basis problem for identities in groups, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 376–384.Google Scholar
Ol’shanskiĭ, A. Yu., On residualing homomorphisms and $G$-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (4) (1993), 365–409.Google Scholar
Robinson, D. J. S., A course in the theory of groups, second ed., Graduate Texts in Mathematics, Volume 80 (Springer-Verlag, New York, 1996).Google Scholar
Serre, J.-P., Arbres, amalgames, ${\mathrm{SL} }_{2} $, (Société Mathématique de France, Paris, 1977), avec un sommaire anglais, rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46.Google Scholar
Strebel, R., Finitely presented soluble groups, in Group Theory, pp. 257–314 (Academic Press, London, 1984).Google Scholar
Suslin, A. A., On the structure of the special linear group over polynomial rings, Math. USSR, Izv. 11 (1977), 221–238 English.CrossRefGoogle Scholar
Tyrer Jones, J. M., Direct products and the Hopf property, J. Aust. Math. Soc. 17 (1974), 174–196. Collection of articles dedicated to the memory of Hanna Neumann, VI.Google Scholar
Vaughan-Lee, M. R., Uncountably many varieties of groups, Bull. Lond. Math. Soc. 2 (1970), 280–286.Google Scholar
Vershik, A. M. and Gordon, E. I., Groups that are locally embeddable in the class of finite groups, Algebra i Analiz 9 (1) (1997), 71–97.Google Scholar