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Degenerations of the hyperbolic space. (English) Zbl 0652.57009
Let \({\mathcal H}^ n(G)\) be the space of discrete faithful representations of a group G in the group of isometries of hyperbolic n-space (the Teichmüller space for \(n=2)\). Morgan and Shalen, motivated by work of Thurston, have shown that \({\mathcal H}^ n(G)\) admits a natural compactification; the new points correspond to isometric actions of G on \({\mathbb{R}}\)-trees (countable increasing unions of metric trees). They use tools from algebraic geometry; in the present nicely written paper a different proof is given remaining in the realm of hyperbolic geometry. An important ingredient is the notion of “convergence of compact metric spaces” due to Gromov.
Reviewer: B.Zimmermann

MSC:
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57S30 Discontinuous groups of transformations
20C99 Representation theory of groups
53C30 Differential geometry of homogeneous manifolds
30F20 Classification theory of Riemann surfaces
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References:
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