# Quelques relations entre propriétés algébriques des groupes de transformation et géométrie des espaces

Abstract : We are interested in (discrete and isometric) actions of a group $\Gamma$ on a measured metric space $X$ and in estimating how these actions separate points. The classical Margulis lemma is a basic result in this subject when $X$ is a simply connected manifold with
negative and bounded curvature. A recent version (due to G. Besson, G. Courtois and S. Gallot) generalises this to the case where $X$ is a
measured metric space with bounded entropy, but it is essentially limited to the case where the group $\Gamma$ is the fundamental group of some manifold with bounded negative curvature and injectivity radius bounded from below. We show that the latter result (and its
geometric corollaries) can be generalized to a larger class ${\cal C}$ of groups (containing word-hyperbolic
groups, free products and malnormal amalgamated products) and to quasi-actions by quasi-isometries (with eventual fixed points) of such groups on a measured metric space of bounded entropy. Applying this result in the case where $X$ is the Cayley graph of a group $G$ which is commensurable to some group $\Gamma \in {\cal C }$, we obtain finiteness results which apply in particular to word-hyperbolic groups
and to fundamental groups of manifolds with bounded diameter. These results aim to understand the questions about the existence of some universal lower bound of the algebraic entropy (for the set of such groups $G$) and about the existence, for any such group, of a generating set with minimal entropy.
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Cited literature [70 references]

https://tel.archives-ouvertes.fr/tel-00011158
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Submitted on : Wednesday, December 7, 2005 - 10:09:24 AM
Last modification on : Wednesday, November 4, 2020 - 2:17:34 PM
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• HAL Id : tel-00011158, version 1

### Citation

Fabio Zuddas. Quelques relations entre propriétés algébriques des groupes de transformation et géométrie des espaces. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2005. Français. ⟨tel-00011158⟩

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