Pour un espace localement symétrique nous définissons une compactification que nous appelons “compactification géodésique”. Elle est construite en ajoutant des points limites dans à certaines géodésiques dans . 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 . De plus 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
For a locally symmetric space , we define a compactification which we call the “geodesic compactification”. It is constructed by adding limit points in to certain geodesics in . The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give for locally symmetric spaces. Moreover, 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 .
Classification : 20G30, 22E40, 58D19, 54A20, 54D35, 31C20
Mots clés : compactifications, espaces localement symétriques, géodésiques, groupes arithmétiques
@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}, pages = {457--559}, publisher = {Association des Annales de l'institut Fourier}, volume = {52}, number = {2}, year = {2002}, doi = {10.5802/aif.1893}, zbl = {1017.53039}, mrnumber = {1906482}, language = {en}, url = {archive.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, Tome 52 (2002) no. 2, pp. 457-559. doi : 10.5802/aif.1893. http://archive.numdam.org/item/AIF_2002__52_2_457_0/
[AR1] A Trace Formula for Reductive Groups I, Duke Math. J., Volume 45 (1978), pp. 911-952 | Article | MR 518111 | Zbl 0499.10032
[AR2] Eisenstein Series and the Trace Formula, Part 1, Proc. Symp. Pure Math., Volume 33 (1979), pp. 253-274 | MR 546601 | Zbl 0431.22016
[BGS] Manifolds of Nonpositive Curvature, Progress in Math., Volume vol. 61, Birkhäuser, Boston, 1985 | MR 823981 | Zbl 0591.53001
[BH] Arithmetic Subgroups of Algebraic Groups, Ann. of Math., Volume 75 (1962), pp. 485-535 | Article | Zbl 0107.14804
[BJ] Compactification of Locally Symmetric Spaces (2000) (Preprint)
[BO1] Introduction aux groupes arithmétiques, Hermann, Paris, 1969 | Zbl 0186.33202
[BO3] Linear Algebraic Groups, Proc. Symp. Pure Math., Volume 9 (1969), pp. 3-19 | MR 204532 | Zbl 0205.50503
[BO4] Reduction theory for arithmetic groups, Proc. Symp. Pure Math., Volume 9 (1969), pp. 20-25 | MR 204533 | Zbl 0213.47201
[BO2] Some Metric Properties of Arithmetic Quotients of Symmetric Spaces and an Extension Theorem, J. Diff. Geom., Volume 6 (1972), pp. 543-560 | MR 338456 | Zbl 0249.32018
[BR] Lectures on Potential Theory, Tata Institute of Fundamental Research (1967)
[BS] Corners and Arithmetic Groups, Comment. Math. Helv., Volume 48 (1973), pp. 436-491 | Article | Zbl 0274.22011
[BT] Groupes réductifs, Publ. Math. IHES, Volume 27 (1965), pp. 55-151 | Numdam | Zbl 0145.17402
[DL] Pure Point Spectrum and Negative Curvature for Noncompact Manifolds, Duke Math. J., Volume 46 (1979), pp. 497-503 | Article | MR 544241 | Zbl 0416.58025
[FR] Harmonic Analysis in Weighted -Spaces, Ann. Sci. École Norm. Sup., Volume 31 (1998), pp. 181-279 | Numdam | MR 1603257 | Zbl 0938.11026
[FRE] Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, Volume 22 (1906), pp. 1-74 | Article | JFM 37.0348.02
[FU] A Poisson Formula for Semi-simple Lie Groups, Ann. Math., Volume 72 (1963), pp. 335-386 | Article | MR 146298 | Zbl 0192.12704
[GHM] Weighted Cohomology, Invent. Math., Volume 116 (1994), pp. 139-213 | Article | MR 1253192 | Zbl 0849.11047
[GJT] Compactifications of Symmetric Spaces, Progress in Math., Volume vol. 156, Birkhäuser, Boston, 1998 | MR 1633171 | Zbl 1053.31006
[GR] Fundamental Domains for Lattices in -Rank 1 Semi-simple Lie Groups, Ann. of Math., Volume 92 (1970), pp. 279-326 | Article | MR 267041 | Zbl 0206.03603
[GR1] Structure métriques pour les variétés riemanniennes, CEDIC, Paris, 1981 | MR 682063 | Zbl 0509.53034
[GR2] Groups of Polynomial Growth and Expanding Groups, IHES, Volume 53 (1981), pp. 53-73 | Numdam | MR 623534 | Zbl 0474.20018
[GR3] Asymptotic Invariants of Infinite Groups, Geometric Group Theory (Sussex, 1991), Volume vol. 2 (1993), pp. 1-295
[GT] Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Volume 224, Springer Verlag, New York, 1977 | MR 473443 | Zbl 0361.35003
[GU] Sojourn ptmr and Asymptotic Properties of the Scattering matrix, RIMS Kyoto Univ., Volume 12 (1977), pp. 69-88 | Article | MR 448453 | Zbl 0381.35064
[HA1] Geometry of Quotient Spaces of by Congruence Subgroups, Math. Ann., Volume 293 (1992), pp. 443-467 | Article | MR 1170519 | Zbl 0769.53033
[HA2] 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 | Article | MR 1317280 | Zbl 0844.53041
[HAD] Les surfaces à courbures opposées et leurs lignes géodésiques, Collected Works, Volume 2, pp. 729-775
[HC] Automorphic Forms on Semisimple Lie Groups, Lecture Notes in Math., Volume vol. 62, Springer-Verlag, 1968 | MR 232893 | Zbl 0186.04702
[HEJ] The Selberg Trace Formula for II, Lecture Notes in Math., Volume vol. 1001, Springer-Verlag, 1983 | MR 711197 | Zbl 0543.10020
[HEL] Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Math., Volume vol. 80, Academic Press, 1978 | MR 514561 | Zbl 0451.53038
[HZ] Boundary Cohomology of Shimura Varieties II. Hodge Theory at the Boundary, Invent. Math., Volume 116 (1994), pp. 243-307 | MR 1253194 | Zbl 0860.11031
[J1] Compactifications of Symmetric and Locally Symmetric Spaces, Geometric Complex Analysis (1996), pp. 297-308 | Zbl 0936.53033
[J2] Metric Compactifications of Locally Symmetric Spaces, Intern. J. of Math., Volume 9 (1998), pp. 465-491 | Article | MR 1635185 | Zbl 0929.32017
[JZ] Scattering matrices and scattering geodesics, Ann. Sci. École Norm. Sup., Volume 34 (2001), pp. 441-469 | Numdam | MR 1839581 | Zbl 1026.53026
[KA] 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 231321 | Zbl 0164.22202
[KE2] General Topology, Graduate Texts in Math., Volume vol. 27, Springer, 1955 | Zbl 0306.54002
[KE1] Wave Propagation, ICM 1994 in Zürich (1994), pp. 106-119 | Zbl 0849.35004
[KT] Fine Convergence and Parabolic Convergence for the Helmholtz Equation and the Heat Equation, Illinois J. Math., Volume 27 (1983), pp. 77-93 | MR 684542 | Zbl 0488.31004
[KU] Topology, Academic Press, 1966 | MR 217751 | Zbl 0158.40802
[LA] On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., Volume vol. 544, Springer-Verlag, 1976 | MR 579181 | Zbl 0332.10018
[LE] Geodesic Rays in Locally Symmetric Spaces, Diff. Geom. Appl., Volume 6 (1996), pp. 55-65 | Article | MR 1384879 | Zbl 0846.53032
[MA] Minimal Positive Harmonic Functions, Trans. Amer. Math. Soc., Volume 49 (1941), pp. 137-172 | Article | JFM 67.0343.03 | MR 3919 | Zbl 0025.33302
[ME] Geometric Scattering Theory, Cambridge University Press, New York, 1995 | MR 1350074 | Zbl 0849.58071
[MU] The Trace Class Conjecture in the Theory of Automorphic Forms, Ann. of Math., Volume 130 (1989), pp. 473-529 | Article | MR 1025165 | Zbl 0701.11019
[MW] Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995 | MR 1361168 | Zbl 0846.11032
[OW2] The Theory of Eisenstein Systems, Pure Appl. Math., Volume vol. 99, Academic Press, 1981 | MR 643242 | Zbl 0489.43009
[OW1] The Selberg Trace Formula II: Partition, Reduction, Truncation, Pacific J. Math., Volume 106 (1983), pp. 307-496 | MR 699915 | Zbl 0515.22013
[SA1] On Representations and Compactifications of Symmetric Spaces, Ann. of Math., Volume 71 (1960), pp. 77-110 | Article | MR 118775 | Zbl 0094.34603
[SA2] On Compactifications of the Quotient Spaces for Arithmetically Defined Discontinuous Groups, Ann. of Math., Volume 72 (1960), pp. 555-580 | Article | MR 170356 | Zbl 0146.04701
[SA] Tilings and Finite Energy Retractions of Locally Symmetric Spaces, Comment. Math. Helv., Volume 72 (1997), pp. 167-202 | Article | MR 1470087 | Zbl 0890.22003
[SE2] 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 88511 | Zbl 0072.08201
[SE1] 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 0197.18002
[SI] Symplectic Geometry, Academic Press, 1964 | MR 164063 | Zbl 0138.31403
[SU1] Disjoint Spheres, Approximation by Imaginary Quadratic Numbers, and the Logarithm Law for Geodesics, Acta Math., Volume 149 (1982), pp. 215-236 | Article | MR 688349 | Zbl 0517.58028
[SU2] Related Aspects of Positivity in Riemannian geometry, J. Diff. Geom., Volume 25 (1987), pp. 327-351 | MR 882827 | Zbl 0615.53029
[TI1] On Buildings and Their Applications, Proc. ICM, Vancouver (1974), pp. 209-220 | Zbl 0336.57009
[TI2] Buildings of Spherical Type and BN-Pairs, Lecture Notes in Math., Volume vol. 386, Springer-Verlag, 1974 | MR 470099 | Zbl 0295.20047
[WI] Scattering States and Wave Operators in the Abstract Theory of Scattering, J. Func. Anal., Volume 12 (1973), pp. 257-274 | Article | MR 344912 | Zbl 0248.47006
[ZI] Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984 | MR 776417 | Zbl 0571.58015
[ZU1] Cohomology of Warped Products and Arithmetic Groups, Invent. Math., Volume 70 (1982), pp. 169-218 | Article | MR 684171 | Zbl 0508.20020
[ZU2] Satake Compactifications, Comment. Math. Helv., Volume 58 (1983), pp. 312-343 | Article | MR 705539 | Zbl 0565.22009