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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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