Compactification via le spectre réel d’espaces des classes de représentation dans SO$(n,1)$

Bouzoubaa, Taoufik

Annales de l'Institut Fourier, Tome 44 (1994) no. 2, pp. 347-385.

Soit $\Gamma$ un groupe de type fini non élémentaire. On note $D^n\left(\Gamma \right)$ l’ensemble des structures hyperboliques de dimension $n$ sur $\Gamma$. $D^n\left(\Gamma \right)$ peut se réaliser comme fermé dans un espace semi-algébrique qui admet une compactification naturelle par le spectre réel. On note ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$ le compactifié via le $\underline{spectre}$ réel de $D^n\left(\Gamma \right)$. L’objet de cet article est de décrire les points ajoutés dans ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$. La compactification obtenue de cette manière permet d’interpréter “les points frontières” comme des représentations de $\Gamma$ dans ${\mathrm{SO}}_F^{+}(n,1)$ où $F(\supset ℝ)$ est un corps réel clos non archimédien. De là on peut retrouver, en construisant un arbre comme quotient de l’espace $n$-hyperbolique sur $F$, une interprétation semblable à celle de Morgan pour les points de sa compactification obtenus comme représentations dans le groupe des isométries d’un $ℝ$-arbre.

Let $\Gamma$ be a finitely generated non-elementary group. We denote the set of all $n$-dimensional hyperbolic structures on $\Gamma$ by $D^n\left(\Gamma \right)$. $D^n\left(\Gamma \right)$ can be realized as a closed subset of a real algebraic set, which has a natural real compactification, denoted by ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$. Our goal here is to describe the boundary points of ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$. We obtain from the boundary points of this compactification certain representations of $\Gamma$ into SO${}_F^{+}(n,1)$, where $F(\supset ℝ)$ 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 ie: as representations into the isometry groups of $ℝ$-trees.

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