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Compactification via the real spectrum of spaces of classes of representations in \(SO(n,1)\). (Compactification via le spectre réel d’espaces des classes de représentation dans \(SO(n,1)\).) (French) Zbl 0803.32015

Let \(\Gamma\) be a finitely generated non-elementary group. We denote the set of all \(n\)-dimensional hyperbolic structures on \(\Gamma\) by \(D^ n (\Gamma)\). \(D^ n (\Gamma)\) can be realized as a closed subset of a real algebraic set, which has a natural real compactification, denoted by \(\overline {D^ n (\Gamma)}^{sp}\). Our goal here is to describe the boundary points of \(\overline {D^ n (\Gamma)}^{sp}\). We obtain from the boundary points of this compactification certain representations of \(\Gamma\) into \(SO^ +_ F(n,1)\), where \(F(\supset \mathbb{R})\) is a non- Archimedean real closed field. By constructing a tree, as quotient space of hyperbolic \(n\)-space over \(F\), we find the same description of boundary points as Morgan’s is: as representations into the isometry groups of \(\mathbb{R}\)-trees.

MSC:

32J05 Compactification of analytic spaces
14P10 Semialgebraic sets and related spaces
57Q15 Triangulating manifolds
20G99 Linear algebraic groups and related topics
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