Geometry of compactifications of locally symmetric spaces

Ji, Lizhen; Macpherson, Robert

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

Pour un espace localement symétrique $M$ nous définissons une compactification $M\cup M\left(\infty \right)$ que nous appelons “compactification géodésique”. Elle est construite en ajoutant des points limites dans $M\left(\infty \right)$ 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 $M\left(\infty \right)$. De plus $M\left(\infty \right)$ 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 $L_2$

For a locally symmetric space $M$, we define a compactification $M\cup M\left(\infty \right)$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M\left(\infty \right)$ 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 $M\left(\infty \right)$ for locally symmetric spaces. Moreover, $M\left(\infty \right)$ 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 $L_2$.

Classification : 20G30, 22E40, 58D19, 54A20, 54D35, 31C20
Keywords: compactifications, locally symmetric spaces, geodesics, arithmetic groups
Mot clés : compactifications, espaces localement symétriques, géodésiques, groupes arithmétiques

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