Geometry of compactifications of locally symmetric spaces

Ji, Lizhen; Macpherson, Robert

Ji, Lizhen; Macpherson, Robert

Annales de l'Institut Fourier, Volume 52 (2002) no. 2, p. 457-559

Abstract
Résumé

For a locally symmetric space $M$ , we define a compactification $M \cup M (\infty)$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M (\infty)$ 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 $M (\infty)$ for locally symmetric spaces. Moreover, $M (\infty)$ 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 $M \cup M (\infty)$ que nous appelons “compactification géodésique”. Elle est construite en ajoutant des points limites dans $M (\infty)$ 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 $M (\infty)$ . De plus $M (\infty)$ 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 $L_{2}$

MR 1906482 | Zbl 1017.53039

DOI : https://doi.org/10.5802/aif.1893
Classification: 20G30, 22E40, 58D19, 54A20, 54D35, 31C20
Keywords: compactifications, locally symmetric spaces, geodesics, arithmetic groups

@article{AIF_2002__52_2_457_0,
     author = {Ji, Lizhen and Macpherson, Robert},
     title = {Geometry of compactifications of locally symmetric spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     pages = {457-559},
     doi = {10.5802/aif.1893},
     zbl = {1017.53039},
     mrnumber = {1906482},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_2_457_0}
}

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. http://www.numdam.org/item/AIF_2002__52_2_457_0/

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