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Topologie de Gromov équivariante, structures hyperboliques et arbres réels. (Equivariant Gromov topology, hyperbolic structures, and \({\mathbb{R}}\)-trees). (French) Zbl 0673.57034
Summary: Les objets que nous étudions sont les espaces métriques munis d’une action par isométrie d’un groupe fixé \(\Gamma\). Nous définissons une “topologie” naturelle sur “l’ensemble” de ces espaces. Nous montrons un critère de compacité séquentielle par des méthodes inspirées des travaux de M. Gromov. Nous utilisons ce critère pour donner une preuve plus courte et plus géométrique de deux théorèmes: celui de M. Culler et J. Morgan sur la compacité de l’espace des arbres réels à petits stabilisateurs d’arêtes; et celui de J. Morgan sur la compacitification de l’espace des structures hyperboliques sur une variété par des arbres réels à petits stabilisateurs d’arêtes.

MSC:
57S30 Discontinuous groups of transformations
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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References:
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