Geometry of compactifications of locally symmetric spaces
[Géométrie des compactifications des espaces localement symétriques]
Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 457-559.

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

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 .

DOI : 10.5802/aif.1893
Classification : 20G30, 22E40, 58D19, 54A20, 54D35, 31C20
Keywords: compactifications, locally symmetric spaces, geodesics, arithmetic groups
Mot clés : compactifications, espaces localement symétriques, géodésiques, groupes arithmétiques
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, Tome 52 (2002) no. 2, pp. 457-559. doi : 10.5802/aif.1893. http://archive.numdam.org/articles/10.5802/aif.1893/

[AR1] J. Arthur A Trace Formula for Reductive Groups I, Duke Math. J., Volume 45 (1978), pp. 911-952 | DOI | MR | Zbl

[AR2] J. Arthur Eisenstein Series and the Trace Formula, Part 1, Proc. Symp. Pure Math., Volume 33 (1979), pp. 253-274 | MR | Zbl

[BGS] W. Ballmann; M. Gromov; V. Schroeder Manifolds of Nonpositive Curvature, Progress in Math., vol. 61, Birkhäuser, Boston, 1985 | MR | Zbl

[BH] A. Borel; Harish-Chandra Arithmetic Subgroups of Algebraic Groups, Ann. of Math., Volume 75 (1962), pp. 485-535 | DOI | Zbl

[BJ] A. Borel; L. Ji Compactification of Locally Symmetric Spaces (2000) (Preprint)

[BO1] A. Borel Introduction aux groupes arithmétiques, Hermann, Paris, 1969 | Zbl

[BO2] A. Borel Some Metric Properties of Arithmetic Quotients of Symmetric Spaces and an Extension Theorem, J. Diff. Geom., Volume 6 (1972), pp. 543-560 | MR | Zbl

[BO3] A. Borel Linear Algebraic Groups, Proc. Symp. Pure Math., Volume 9 (1969), pp. 3-19 | MR | Zbl

[BO4] A. Borel Reduction theory for arithmetic groups, Proc. Symp. Pure Math., Volume 9 (1969), pp. 20-25 | MR | Zbl

[BR] M. Brelot Lectures on Potential Theory, Tata Institute of Fundamental Research (1967)

[BS] A. Borel; J.-P. Serre Corners and Arithmetic Groups, Comment. Math. Helv., Volume 48 (1973), pp. 436-491 | DOI | Zbl

[BT] A. Borel; T. Tits Groupes réductifs, Publ. Math. IHES, Volume 27 (1965), pp. 55-151 | Numdam | Zbl

[DL] H. Donnelly; P. Li Pure Point Spectrum and Negative Curvature for Noncompact Manifolds, Duke Math. J., Volume 46 (1979), pp. 497-503 | DOI | MR | Zbl

[FR] J. Franke Harmonic Analysis in Weighted L 2 -Spaces, Ann. Sci. École Norm. Sup., Volume 31 (1998), pp. 181-279 | Numdam | MR | Zbl

[FRE] M. Fréchet Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, Volume 22 (1906), pp. 1-74 | DOI | JFM

[FU] H. Furstenberg A Poisson Formula for Semi-simple Lie Groups, Ann. Math., Volume 72 (1963), pp. 335-386 | DOI | MR | Zbl

[GHM] M. Goresky; G. Harder; R. MacPherson Weighted Cohomology, Invent. Math., Volume 116 (1994), pp. 139-213 | DOI | MR | Zbl

[GJT] Y. Guivarch; L. Ji; J.C. Taylor Compactifications of Symmetric Spaces, Progress in Math., vol. 156, Birkhäuser, Boston, 1998 | MR | Zbl

[GR] H. Garland; M.S. Raghunathan Fundamental Domains for Lattices in ()-Rank 1 Semi-simple Lie Groups, Ann. of Math., Volume 92 (1970), pp. 279-326 | DOI | MR | Zbl

[GR1] M. Gromov Structure métriques pour les variétés riemanniennes, CEDIC, Paris, 1981 | MR | Zbl

[GR2] M. Gromov Groups of Polynomial Growth and Expanding Groups, IHES, Volume 53 (1981), pp. 53-73 | Numdam | MR | Zbl

[GR3] M. Gromov Asymptotic Invariants of Infinite Groups, Geometric Group Theory (Sussex, 1991), Volume vol. 2 (1993), pp. 1-295

[GT] D. Gilbarg; N. Trudinger Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, 224, Springer Verlag, New York, 1977 | MR | Zbl

[GU] V. Guillemin Sojourn ptmr and Asymptotic Properties of the Scattering matrix, RIMS Kyoto Univ., Volume 12 (1977), pp. 69-88 | DOI | MR | Zbl

[HA1] T. Hattori Geometry of Quotient Spaces of SO(3)SL(3,) by Congruence Subgroups, Math. Ann., Volume 293 (1992), pp. 443-467 | DOI | MR | Zbl

[HA2] T. Hattori Collapsing of Quotient Spaces of $\hbox{SO}(n)\backslash \hbox{SL}(n,\Bbb R) at Infinity, J. Math. Soc. Japan, Volume 47 (1995), pp. 193-225 | DOI | MR | Zbl

[HAD] M. Hadamard Les surfaces à courbures opposées et leurs lignes géodésiques, Collected Works, Volume 2, pp. 729-775

[HC] Harish-Chandra Automorphic Forms on Semisimple Lie Groups, Lecture Notes in Math., vol. 62, Springer-Verlag, 1968 | MR | Zbl

[HEJ] D. Hejhal The Selberg Trace Formula for PSL(2,) II, Lecture Notes in Math., vol. 1001, Springer-Verlag, 1983 | MR | Zbl

[HEL] S. Helgason Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Math., vol. 80, Academic Press, 1978 | MR | Zbl

[HZ] M. Harris; S. Zucker Boundary Cohomology of Shimura Varieties II. Hodge Theory at the Boundary, Invent. Math., Volume 116 (1994), pp. 243-307 | MR | Zbl

[J1] L. Ji; Noguich et al. (eds.) Compactifications of Symmetric and Locally Symmetric Spaces, Geometric Complex Analysis (1996), pp. 297-308 | Zbl

[J2] L. Ji Metric Compactifications of Locally Symmetric Spaces, Intern. J. of Math., Volume 9 (1998), pp. 465-491 | DOI | MR | Zbl

[JZ] L. Ji; M. Zworski Scattering matrices and scattering geodesics, Ann. Sci. École Norm. Sup., Volume 34 (2001), pp. 441-469 | Numdam | MR | Zbl

[KA] F.I. Karpelevic The Geometry of Geodesics and the Eigenfunctions of the Beltrami-Laplace Operator on Symmetric Spaces, Trans. Moscow Math. Soc., Volume 14 (1965), pp. 51-199 | MR | Zbl

[KE1] J. Keller Wave Propagation, ICM 1994 in Zürich (1994), pp. 106-119 | Zbl

[KE2] J.L. Kelly General Topology, Graduate Texts in Math., vol. 27, Springer, 1955 | Zbl

[KT] A. Koranyi; J.C. Taylor Fine Convergence and Parabolic Convergence for the Helmholtz Equation and the Heat Equation, Illinois J. Math., Volume 27 (1983), pp. 77-93 | MR | Zbl

[KU] K. Kuratowski Topology, Academic Press, 1966 | MR | Zbl

[LA] R. Langlands On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., vol. 544, Springer-Verlag, 1976 | MR | Zbl

[LE] E. Leuzinger Geodesic Rays in Locally Symmetric Spaces, Diff. Geom. Appl., Volume 6 (1996), pp. 55-65 | DOI | MR | Zbl

[MA] R.S. Martin Minimal Positive Harmonic Functions, Trans. Amer. Math. Soc., Volume 49 (1941), pp. 137-172 | DOI | JFM | MR | Zbl

[ME] R. Melrose Geometric Scattering Theory, Cambridge University Press, New York, 1995 | MR | Zbl

[MU] W. Müller The Trace Class Conjecture in the Theory of Automorphic Forms, Ann. of Math., Volume 130 (1989), pp. 473-529 | DOI | MR | Zbl

[MW] C. Moeglin; J.L. Waldspurger Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995 | MR | Zbl

[OW1] M.S. Osborne; G. Warner The Selberg Trace Formula II: Partition, Reduction, Truncation, Pacific J. Math., Volume 106 (1983), pp. 307-496 | MR | Zbl

[OW2] M.S. Osborne; G. Warner The Theory of Eisenstein Systems, Pure Appl. Math., vol. 99, Academic Press, 1981 | MR | Zbl

[SA] L. Saper Tilings and Finite Energy Retractions of Locally Symmetric Spaces, Comment. Math. Helv., Volume 72 (1997), pp. 167-202 | DOI | MR | Zbl

[SA1] I. Satake On Representations and Compactifications of Symmetric Spaces, Ann. of Math., Volume 71 (1960), pp. 77-110 | DOI | MR | Zbl

[SA2] I. Satake On Compactifications of the Quotient Spaces for Arithmetically Defined Discontinuous Groups, Ann. of Math., Volume 72 (1960), pp. 555-580 | DOI | MR | Zbl

[SE1] A. Selberg Recent Developments in the Theory of Discontinuous Groups of Motions of Symmetric Spaces (Lecture Notes in Math.), Volume vol. 118 (1968), pp. 99-120 | Zbl

[SE2] A. Selberg Harmonic Analysis and Discontinuous Groups in Weakly Riemannian Symmetric Spaces with Applications to Dirichlet Series, J. Ind. Math. Soc., Volume 20 (1956), pp. 47-87 | MR | Zbl

[SI] C.L. Siegel Symplectic Geometry, Academic Press, 1964 | MR | Zbl

[SU1] D. Sullivan Disjoint Spheres, Approximation by Imaginary Quadratic Numbers, and the Logarithm Law for Geodesics, Acta Math., Volume 149 (1982), pp. 215-236 | DOI | MR | Zbl

[SU2] D. Sullivan Related Aspects of Positivity in Riemannian geometry, J. Diff. Geom., Volume 25 (1987), pp. 327-351 | MR | Zbl

[TI1] J. Tits On Buildings and Their Applications, Proc. ICM, Vancouver (1974), pp. 209-220 | Zbl

[TI2] J. Tits Buildings of Spherical Type and BN-Pairs, Lecture Notes in Math., vol. 386, Springer-Verlag, 1974 | MR | Zbl

[WI] C. Wilcox Scattering States and Wave Operators in the Abstract Theory of Scattering, J. Func. Anal., Volume 12 (1973), pp. 257-274 | DOI | MR | Zbl

[ZI] R. Zimmer Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984 | MR | Zbl

[ZU1] S. Zucker L 2 Cohomology of Warped Products and Arithmetic Groups, Invent. Math., Volume 70 (1982), pp. 169-218 | DOI | MR | Zbl

[ZU2] S. Zucker Satake Compactifications, Comment. Math. Helv., Volume 58 (1983), pp. 312-343 | DOI | MR | Zbl

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