On the Cantor-Bendixson rank of metabelian groups
Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 593-618.

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence (G n ) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank ω n .

On étudie le rang de Cantor-Bendixson des groupes métabéliens ou virtuellement métabéliens dans l’espace des groupes marqués, et on exhibe notamment une suite (G n ) de groupes virtuellement métabéliens de présentation finie à deux générateurs, de rang de Cantor-Bendixson égal à ω n .

DOI: 10.5802/aif.2623
Classification: 20E15, 13E05, 20F05, 20F16, 57M07
Keywords: Metabelian groups, space of marked groups, Cantor-Bendixson analysis, Bieri-Strebel invariant, lattice of subgroups
Mot clés : groupes métabéliens, espace des groupes marqués, analyse de Cantor-Bendixson, invariant de Bieri-Strebel, treillis des sous-groupes
Cornulier, Yves 1

1 IRMAR Campus de Beaulieu 35042 Rennes Cedex (France)
@article{AIF_2011__61_2_593_0,
     author = {Cornulier, Yves},
     title = {On the {Cantor-Bendixson} rank of metabelian groups},
     journal = {Annales de l'Institut Fourier},
     pages = {593--618},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     doi = {10.5802/aif.2623},
     zbl = {1238.20049},
     mrnumber = {2895067},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2623/}
}
TY  - JOUR
AU  - Cornulier, Yves
TI  - On the Cantor-Bendixson rank of metabelian groups
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 593
EP  - 618
VL  - 61
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2623/
DO  - 10.5802/aif.2623
LA  - en
ID  - AIF_2011__61_2_593_0
ER  - 
%0 Journal Article
%A Cornulier, Yves
%T On the Cantor-Bendixson rank of metabelian groups
%J Annales de l'Institut Fourier
%D 2011
%P 593-618
%V 61
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2623/
%R 10.5802/aif.2623
%G en
%F AIF_2011__61_2_593_0
Cornulier, Yves. On the Cantor-Bendixson rank of metabelian groups. Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 593-618. doi : 10.5802/aif.2623. http://www.numdam.org/articles/10.5802/aif.2623/

[1] Abels, H. An example of a finitely presented solvable group, Homological group theory (Proc. Sympos., Durham, 1977), Volume 36, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 1979, pp. 205-211 | MR | Zbl

[2] Bass, H. Descending chains and the Krull ordinal of commutative Noetherian rings, J. Pure Appl. Algebra, Volume 1 (1971), pp. 347-360 | DOI | MR | Zbl

[3] Baumslag, G.; Solitar, D. Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc., Volume 68 (1962), pp. 199-201 | DOI | MR | Zbl

[4] Bieri, R.; Strebel, R. Valuations and finitely presented metabelian groups, Proc. London Math. Soc., Volume 41 (1980) no. 3, pp. 439-464 | DOI | MR | Zbl

[5] Bieri, R.; Strebel, R. A geometric invariant for modules over an abelian group, J. Reine Angew. Math., Volume 322 (1981), pp. 170-189 | MR | Zbl

[6] Chabauty, C. Limite d’ensembles et géométrie des nombres, Bull. Soc. Math. France, Volume 78 (1950), pp. 143-151 | Numdam | MR | Zbl

[7] Champetier, C. L’espace des groupes de type fini, Topology, Volume 39 (2000) no. 4, pp. 657-680 | DOI | MR | Zbl

[8] Champetier, C.; Guirardel, V. Limit groups as limits of free groups, Israel J. Math., Volume 146 (2005), pp. 1-75 | DOI | MR | Zbl

[9] Cornulier, Y. Finitely presented wreath products and double coset decompositions, Geom. Dedicata, Volume 122 (2006), pp. 89-108 | DOI | MR | Zbl

[10] Cornulier, Y. The space of finitely generated rings, Internat. J. Algebra Comput., Volume 19 (2009) no. 3, pp. 373-382 | DOI | MR | Zbl

[11] Cornulier, Y.; Guyot, L.; Pitsch, W. On the isolated points in the space of groups, J. Algebra, Volume 307 (2007) no. 7, pp. 254-277 | DOI | MR | Zbl

[12] Grigorchuk, R. Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., Volume 48 (1984) no. 5, pp. 939-985 | MR | Zbl

[13] Gulliksen, H. A theory of length for Noetherian modules, J. Pure Appl. Algebra, Volume 3 (1973), pp. 159-170 | DOI | MR | Zbl

[14] Hall, P. Finiteness conditions for soluble groups, Proc. London Math. Soc. i (1954) no. 16, pp. 419-436 | DOI | MR | Zbl

[15] Hall, P. On the finiteness of certain soluble groups, Proc. London Math. Soc., Volume 9 (1959) no. 3, pp. 595-622 | DOI | MR | Zbl

[16] Magnus, W. On a theorem of Marshall Hall, Annals of Math., Volume 40 (1939) no. 4, pp. 764-768 | DOI | JFM | MR | Zbl

[17] McConnell, J.; Robson, J. Noncommutative noetherian rings, Grad. Stud. Math., 30, Amer. Math. Soc. Providence, 2001 | MR | Zbl

[18] Olshanskii, A. On residualing homomorphisms and G-subgroups of hyperbolic groups, Internat. J. Algebra Comput., Volume 3 (1993) no. 4, pp. 365-409 | DOI | MR | Zbl

[19] Roseblade, J. Group rings of polycyclic groups, J. Pure Appl. Algebra, Volume 3 (1973), pp. 307-328 | DOI | MR | Zbl

[20] Serre, J.-P. Cours d’arithmétique, Presses Universitaires de France, 1970 | MR | Zbl

[21] Sierpiński, W. Cardinal and ordinal numbers, Second revised edition. Monografie Matematyczne, 34, Państowe Wydawnictwo Naukowe, Warsaw, 1965 | MR | Zbl

Cited by Sources: