Compactification via le spectre réel d’espaces des classes de représentation dans SO(n,1)
Annales de l'Institut Fourier, Volume 44 (1994) no. 2, p. 347-385

Let Γ be a finitely generated non-elementary group. We denote the set of all n-dimensional hyperbolic structures on Γ by D n (Γ). D n (Γ) can be realized as a closed subset of a real algebraic set, which has a natural real compactification, denoted by D n (Γ) ¯ sp . Our goal here is to describe the boundary points of D n (Γ) ¯ sp . We obtain from the boundary points of this compactification certain representations of Γ into SO F + (n,1), where F() 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 -trees.

Soit Γ un groupe de type fini non élémentaire. On note D n (Γ) l’ensemble des structures hyperboliques de dimension n sur Γ. D n (Γ) peut se réaliser comme fermé dans un espace semi-algébrique qui admet une compactification naturelle par le spectre réel. On note D n (Γ) ¯ sp le compactifié via le spectre ̲ réel de D n (Γ). L’objet de cet article est de décrire les points ajoutés dans D n (Γ) ¯ sp . La compactification obtenue de cette manière permet d’interpréter “les points frontières” comme des représentations de Γ dans SO F + (n,1)F() 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 -arbre.

@article{AIF_1994__44_2_347_0,
     author = {Bouzoubaa, Taoufik},
     title = {Compactification via le spectre r\'eel d'espaces des classes de repr\'esentation dans SO$(n,1)$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {2},
     year = {1994},
     pages = {347-385},
     doi = {10.5802/aif.1401},
     zbl = {0803.32015},
     mrnumber = {96e:14065},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1994__44_2_347_0}
}
Bouzoubaa, Taoufik. Compactification via le spectre réel d’espaces des classes de représentation dans SO$(n,1)$. Annales de l'Institut Fourier, Volume 44 (1994) no. 2, pp. 347-385. doi : 10.5802/aif.1401. http://www.numdam.org/item/AIF_1994__44_2_347_0/

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