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), p. 347-385 / Harvested from Numdam

Résumé

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

Abstract

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 $\overline{D^n(\Gamma )}^{\rm sp}$. Our goal here is to describe the boundary points of $\overline{D^n(\Gamma )}^{\rm sp}$. We obtain from the boundary points of this compactification certain representations of $\Gamma $ into SO$_F^+(n,1)$, where $F(\supset {\Bbb R})$ 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 ${\Bbb R}$-trees.

Détail

Publié le : 1994-01-01

MR 96e:14065 | Zbl 0803.32015

DOI : https://doi.org/10.5802/aif.1401

Bibliographie

[AB] R. Alperin, H. Bass, Length functions of group actions on Λ-Trees, Combinatorial Group Theory and Topology (S.M. Gersten and J.R. Stallings, eds.), Ann. of Math. Studies, vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, 265-378. | MR 89c:20057 | Zbl 0978.20500

[Br1] G.-W. Brumfiel, The real spectrum compactification of Teichmüller space, Contemp. Math., 74 (1988), 51-75. | MR 90b:32042 | Zbl 0662.32022

[Br2] G.-W. Brumfiel, The tree of non archimediean hyperbolic plane, Contemp. Math., 74 (1988), 83-106. | MR 89m:51024 | Zbl 0657.51013

[BCR] J. Bochnak, M. Coste, M.-F. Roy, Géométrie algébrique réelle, Springer-Verlag. | Zbl 0633.14016

[CC] M. Coste, M. Carral, Normal spectral spaces and their dimensions, Journal of Pure and applied Algebra, 30 (1983), 227-235. | MR 85c:14002 | Zbl 0525.14015

[FLP] A. Fathi, F. Laudenbach, V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque, (1979), 66-67. | MR 82m:57003 | Zbl 0406.00016

[He] H. Helling, Diskrete untergruppen von SL2 (ℝ), Invent. Math., 17 (1972), 217-229. | MR 48 #2427 | Zbl 0239.22015

[M1] J.-W. Morgan, Group actions on trees and compactification of the space of classes of SO(n, 1) representations, Topology, 25 (1986), 1-33. | MR 87h:20062 | Zbl 0595.57030

[M2] J.-W. Morgan, Λ-Trees and their applications., Bulletin (New series) of the American Mathematical Society, 26, number 1 (January 1992). | MR 92e:20017 | Zbl 0767.05054

[MS] J.-W. Morgan, P.-B. Shalen, Valuation trees, and degenerations of hyperbolic structures, Annals of Math., 120 (1984), 401-476. | Zbl 0583.57005

[P] C. Procesi, The invariant theory of n ˟ n matrices, Advances in Mathematics, 19 (1976), 306-381. | MR 54 #7512 | Zbl 0331.15021

[SS] M. Seppälä, T. Sorvali, Affine coordinates for Teichmüller spaces, Ann. Math., 284 (1989), 165-176. | Zbl 0649.32018

[S] T. Springer, Invariant Theory, Lecture notes in Mathematics, 585, Springer-Verlag, New York, 1977. | MR 56 #5740 | Zbl 0346.20020

[Sh] M. Shiota, Piecewise Linearisation of subanalytic fonctions II, Lecture notes in Mathematics, 1420, 247-307. | Zbl 0716.32006

[ShY] M. Shiota, M. Yokoi, Triangulations of Subanalytic Sets and Locally Subanalytic Manifolds, RIMS-454. | Zbl 0527.57014

[Wi] N.-J. Wielenberg, Discrete Mœbius groups : Fondamental polyhedra and convergence, American Journal of Mathematics, vol. 99, n° 4, p. 861-877. | MR 57 #16579 | Zbl 0373.57024