Estimations de dimensions de Minkowski dans l’espace des groupes marqués
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 1, p. 107-124

In this article we show that the space of marked groups is a closed subspace of a Cantor space whith infinite Hausdorff dimension. We prove that the Minkowski dimension of this space is infinite by exhibiting subsets of marked groups with small cancellation the dimension of which are arbitrarly large. We give estimates of the Minkowski dimensions of subsets of marked groups with one relator. Eventually, we prove that the Minkowski dimensions of the subspace of abelian marked groups and a Cantor space defined by Grigorchuk are zero.

Dans cet article, on montre que l’espace des groupes marqués est un sous-espace fermé d’un ensemble de Cantor dont la dimension de Hausdorff est infinie. On prouve que la dimension de Minkowski de cet espace est infinie en exhibant des sous-ensembles de groupes marqués à petite simplification dont les dimensions de Minkowski sont arbitrairement grandes. On donne une estimation des dimensions de Minkowski de sous-espaces de groupes à un relateur. On démontre enfin que les dimensions de Minkowski du sous-espace des groupes commutatifs marqués et d’un ensemble de Cantor défini par Grigorchuk sont nulles.

@article{AFST_2007_6_16_1_107_0,
     author = {Guyot, Luc},
     title = {Estimations de dimensions de Minkowski dans l'espace des groupes marqu\'es},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 16},
     number = {1},
     year = {2007},
     pages = {107-124},
     doi = {10.5802/afst.1141},
     mrnumber = {2325594},
     zbl = {pre05247241},
     language = {fr},
     url = {http://www.numdam.org/item/AFST_2007_6_16_1_107_0}
}
Guyot, Luc. Estimations de dimensions de Minkowski dans l’espace des groupes marqués. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 1, pp. 107-124. doi : 10.5802/afst.1141. http://www.numdam.org/item/AFST_2007_6_16_1_107_0/

[1] Arzhantseva (G. N.), Ol’shanskii (A. Yu.).— Generality of the class of groups in which subgroups with a lesser number of generators are free. (Russian) Mat. Zametki 59, No. 4, p. 489-496 (1996), 638 ; translation in Math. Notes 59, no. 3-4, p. 350-355 (1996). | MR 1445193 | Zbl 0877.20021

[2] Champetier (C.).— L’espace des groupes de type fini, Topology 39, p. 657-680 (2000). | MR 1760424 | Zbl 0959.20041

[3] Champetier (C.), Guirardel (V.).— Limit groups as limits of free groups. Israel J. Math. 146, p. 1-75 (2005). | MR 2151593 | Zbl 02174701

[4] de la Harpe (P.).— Topics in geometric group theory, Chicago Lectures in Mathematics, The University of Chicago Press (2000). | MR 1786869 | Zbl 0965.20025

[5] Falconer (K.).— Techniques in fractal geometry, John Wiley and sons (1997). | MR 1449135 | Zbl 0869.28003

[6] Grigorchuk (R. I.).— Degrees of growth of finiteley generated groups and the theory of invariant means, Math USSR Iszvestiya 25 No. 2 (1985). | MR 764305 | Zbl 0583.20023

[7] Grigorchuk (R. I.).— An example of finitely presented amenable group not belonging to the class EG, Matematicheskiĭ Sbornik 189, p. 75-95 (1998). | MR 1616436 | Zbl 0931.43003

[8] Gromov (M.).— Hyperbolic groups, Essays in Group Theory, ed. S. M. Gersten, MSRI series, Vol. 8, Springer-Verlag, p. 75-263 (1987). | MR 919829 | Zbl 0634.20015

[9] Lyndon (R.C.), Schupp (P.E.).— Combinatorial Group Theory, Springer (1977). | MR 577064 | Zbl 0368.20023

[10] Pertti (M.).— Geometry of sets and measures in euclidean spaces, fractals and rectifiability. Cambridge Studies in Advanced Mathematics, Cambridge University Press (1995). | MR 1333890 | Zbl 0819.28004

[11] Greendlinger (M.).— An analogue of a theorem of Magnus. Arch. Math 12, p. 94-96 (1961). | MR 132099 | Zbl 0103.25603