Geometry of compactifications of locally symmetric spaces
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 457-559.

For a locally symmetric space M, we define a compactification MM() which we call the “geodesic compactification”. It is constructed by adding limit points in M() to certain geodesics in M. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give M() for locally symmetric spaces. Moreover, M() has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in the harmonic analysis of the locally symmetric space:1) it is the minimal Martin compactification for negative eigenvalues of the Laplacian, and 2) it can be used to parameterize the eigenfunctions of the Laplacian in continuous spectrum on L 2 .

Pour un espace localement symétrique M nous définissons une compactification MM() que nous appelons “compactification géodésique”. Elle est construite en ajoutant des points limites dans M() à certaines géodésiques dans M. La compactification géodésique apparaî t dans d’autres cas. Les constructions générales de Gromov permettent, dans le cas des espaces symétriques, d’identifier le bord de la compactification de Gromov avec M(). De plus M() se construit naturellement avec la théorie des groupes en utilisant l’immeuble de Tits. La compactification géodésique joue deux rôles fondamentaux dans l’analyse harmonique de l’espace localement symétrique : 1) c’est la compactification de Martin minimale pour les valeurs négatives du laplacien et 2) elle peut être utilisée pour paramétrer les valeurs propres du laplacien dans le spectre continu sur L 2

DOI: 10.5802/aif.1893
Classification: 20G30,  22E40,  58D19,  54A20,  54D35,  31C20
Keywords: compactifications, locally symmetric spaces, geodesics, arithmetic groups
Ji, Lizhen 1; Macpherson, Robert 2

1 University of Michigan, Department of Mathematics, Ann Arbor MI 48109-1003 (USA)
2 Institute for Advanced Study, School of Mathematics, Princeton NJ 08540 (USA)
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Ji, Lizhen; Macpherson, Robert. Geometry of compactifications of locally symmetric spaces. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 457-559. doi : 10.5802/aif.1893. https://aif.centre-mersenne.org/articles/10.5802/aif.1893/

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