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Group completions and Fürstenberg boundaries: rank one. (English) Zbl 0567.22005

From the authors introduction: ”In this paper we show that if G is a noncompact, connected semisimple Lie group with finite center and R-rank one and \(\Lambda\) is a uniform lattice of G (that is, a discrete subgroup of G so that \(G/\Lambda\) is compact), then there is a \(\Lambda\)-equivariant homeomorphism \(\phi: {\bar\Lambda}\to B(G)''\) (=Fürstenberg maximal boundary).
”The proof follows the main proof in [Invent. Math. 57, 205-218 (1980; Zbl 0428.20022)], and uses the classification of noncompact, Riemannian symmetric spaces with R-rank one and negative curvature. In § 1 we recall the construction of \({\bar\Lambda}\) and some properties of it. In § 2 we give models for the noncompact, Riemannian symmetric spaces with R-rank one and negative curvature. In § 3 we give the proof and point out that this construction cannot always work if G has rank greater than one.”
Reviewer: H.Leptin

MSC:

22E40 Discrete subgroups of Lie groups

Citations:

Zbl 0428.20022
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References:

[1] W. J. Floyd, Group completions and limit sets of Kleinian groups , Invent. Math. 57 (1980), no. 3, 205-218. · Zbl 0428.20022 · doi:10.1007/BF01418926
[2] H. Furstenberg, A Poisson formula for semi-simple Lie groups , Ann. of Math. (2) 77 (1963), 335-386. JSTOR: · Zbl 0192.12704 · doi:10.2307/1970220
[3] H. Furstenberg, Boundaries of Lie groups and discrete subgroups , Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 301-306. · Zbl 0244.22010
[4] G. D. Mostow, Strong rigidity of locally symmetric spaces , Princeton University Press, Princeton, N.J., 1973. · Zbl 0265.53039
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