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- Série 6 : Tome 16 (2007)

- no. 1

- p. 107-124

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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.

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.

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Luc Guyot 1

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