Compactification via the real spectrum of spaces of classes of representations in \(SO(n,1)\) (Q1337028)

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.