Geometry of compactifications of locally symmetric spaces

Ji, Lizhen; Macpherson, Robert

doi:10.5802/aif.1893

Geometry of compactifications of locally symmetric spaces [ Géométrie des compactifications des espaces localement symétriques ]

Ji, Lizhen ; Macpherson, Robert

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

Résumé
Abstract

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 $L_{2}$

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 $L_{2}$ .

MR 1906482 | Zbl 1017.53039

DOI : https://doi.org/10.5802/aif.1893
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/

Bibliographie

[AR1] J. Arthur A Trace Formula for Reductive Groups I, Duke Math. J., Volume 45 (1978), pp. 911-952 | Article | MR 518111 | Zbl 0499.10032

[AR2] J. Arthur Eisenstein Series and the Trace Formula, Part 1, Proc. Symp. Pure Math., Volume 33 (1979), pp. 253-274 | MR 546601 | Zbl 0431.22016

[BGS] W. Ballmann; M. Gromov; V. Schroeder Manifolds of Nonpositive Curvature, Progress in Math., Volume vol. 61, Birkhäuser, Boston, 1985 | MR 823981 | Zbl 0591.53001

[BH] A. Borel; Harish-Chandra Arithmetic Subgroups of Algebraic Groups, Ann. of Math., Volume 75 (1962), pp. 485-535 | Article | Zbl 0107.14804

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

[BO1] A. Borel Introduction aux groupes arithmétiques, Hermann, Paris, 1969 | Zbl 0186.33202

[BO3] A. Borel Linear Algebraic Groups, Proc. Symp. Pure Math., Volume 9 (1969), pp. 3-19 | MR 204532 | Zbl 0205.50503

[BO4] A. Borel Reduction theory for arithmetic groups, Proc. Symp. Pure Math., Volume 9 (1969), pp. 20-25 | MR 204533 | Zbl 0213.47201

[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 338456 | Zbl 0249.32018

[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 | Article | Zbl 0274.22011

[BT] A. Borel; T. Tits Groupes réductifs, Publ. Math. IHES, Volume 27 (1965), pp. 55-151 | Numdam | Zbl 0145.17402

[DL] H. Donnelly; P. Li 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] J. Franke Harmonic Analysis in Weighted $L_{2}$ -Spaces, Ann. Sci. École Norm. Sup., Volume 31 (1998), pp. 181-279 | Numdam | MR 1603257 | Zbl 0938.11026

[FRE] M. Fréchet Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, Volume 22 (1906), pp. 1-74 | Article | JFM 37.0348.02

[FU] H. Furstenberg A Poisson Formula for Semi-simple Lie Groups, Ann. Math., Volume 72 (1963), pp. 335-386 | Article | MR 146298 | Zbl 0192.12704

[GHM] M. Goresky; G. Harder; R. MacPherson Weighted Cohomology, Invent. Math., Volume 116 (1994), pp. 139-213 | Article | MR 1253192 | Zbl 0849.11047

[GJT] Y. Guivarch; L. Ji; J.C. Taylor Compactifications of Symmetric Spaces, Progress in Math., Volume vol. 156, Birkhäuser, Boston, 1998 | MR 1633171 | Zbl 1053.31006

[GR] H. Garland; M.S. Raghunathan Fundamental Domains for Lattices in $(ℝ)$ -Rank 1 Semi-simple Lie Groups, Ann. of Math., Volume 92 (1970), pp. 279-326 | Article | MR 267041 | Zbl 0206.03603

[GR1] M. Gromov Structure métriques pour les variétés riemanniennes, CEDIC, Paris, 1981 | MR 682063 | Zbl 0509.53034

[GR2] M. Gromov Groups of Polynomial Growth and Expanding Groups, IHES, Volume 53 (1981), pp. 53-73 | Numdam | MR 623534 | Zbl 0474.20018

[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, Volume 224, Springer Verlag, New York, 1977 | MR 473443 | Zbl 0361.35003

[GU] V. Guillemin 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] T. Hattori Geometry of Quotient Spaces of $SO (3) ∖ SL (3, ℝ)$ by Congruence Subgroups, Math. Ann., Volume 293 (1992), pp. 443-467 | Article | MR 1170519 | Zbl 0769.53033

[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 | Article | MR 1317280 | Zbl 0844.53041

[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., Volume vol. 62, Springer-Verlag, 1968 | MR 232893 | Zbl 0186.04702

[HEJ] D. Hejhal The Selberg Trace Formula for $PSL (2, ℝ)$ II, Lecture Notes in Math., Volume vol. 1001, Springer-Verlag, 1983 | MR 711197 | Zbl 0543.10020

[HEL] S. Helgason Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Math., Volume vol. 80, Academic Press, 1978 | MR 514561 | Zbl 0451.53038

[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 1253194 | Zbl 0860.11031

[J1] L. Ji; Noguich et al. (eds.) Compactifications of Symmetric and Locally Symmetric Spaces, Geometric Complex Analysis (1996), pp. 297-308 | Zbl 0936.53033

[J2] L. Ji Metric Compactifications of Locally Symmetric Spaces, Intern. J. of Math., Volume 9 (1998), pp. 465-491 | Article | MR 1635185 | Zbl 0929.32017

[JZ] L. Ji; M. Zworski Scattering matrices and scattering geodesics, Ann. Sci. École Norm. Sup., Volume 34 (2001), pp. 441-469 | Numdam | MR 1839581 | Zbl 1026.53026

[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 231321 | Zbl 0164.22202

[KE2] J.L. Kelly General Topology, Graduate Texts in Math., Volume vol. 27, Springer, 1955 | Zbl 0306.54002

[KE1] J. Keller Wave Propagation, ICM 1994 in Zürich (1994), pp. 106-119 | Zbl 0849.35004

[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 684542 | Zbl 0488.31004

[KU] K. Kuratowski Topology, Academic Press, 1966 | MR 217751 | Zbl 0158.40802

[LA] R. Langlands On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., Volume vol. 544, Springer-Verlag, 1976 | MR 579181 | Zbl 0332.10018

[LE] E. Leuzinger Geodesic Rays in Locally Symmetric Spaces, Diff. Geom. Appl., Volume 6 (1996), pp. 55-65 | Article | MR 1384879 | Zbl 0846.53032

[MA] R.S. Martin 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] R. Melrose Geometric Scattering Theory, Cambridge University Press, New York, 1995 | MR 1350074 | Zbl 0849.58071

[MU] W. Müller 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] C. Moeglin; J.L. Waldspurger Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995 | MR 1361168 | Zbl 0846.11032

[OW2] M.S. Osborne; G. Warner The Theory of Eisenstein Systems, Pure Appl. Math., Volume vol. 99, Academic Press, 1981 | MR 643242 | Zbl 0489.43009

[OW1] M.S. Osborne; G. Warner The Selberg Trace Formula II: Partition, Reduction, Truncation, Pacific J. Math., Volume 106 (1983), pp. 307-496 | MR 699915 | Zbl 0515.22013

[SA1] I. Satake On Representations and Compactifications of Symmetric Spaces, Ann. of Math., Volume 71 (1960), pp. 77-110 | Article | MR 118775 | Zbl 0094.34603

[SA2] I. Satake 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] L. Saper 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] 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 88511 | Zbl 0072.08201

[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 0197.18002

[SI] C.L. Siegel Symplectic Geometry, Academic Press, 1964 | MR 164063 | Zbl 0138.31403

[SU1] D. Sullivan 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] D. Sullivan Related Aspects of Positivity in Riemannian geometry, J. Diff. Geom., Volume 25 (1987), pp. 327-351 | MR 882827 | Zbl 0615.53029

[TI1] J. Tits On Buildings and Their Applications, Proc. ICM, Vancouver (1974), pp. 209-220 | Zbl 0336.57009

[TI2] J. Tits Buildings of Spherical Type and BN-Pairs, Lecture Notes in Math., Volume vol. 386, Springer-Verlag, 1974 | MR 470099 | Zbl 0295.20047

[WI] C. Wilcox 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] R. Zimmer Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984 | MR 776417 | Zbl 0571.58015

[ZU1] S. Zucker $L_{2}$ Cohomology of Warped Products and Arithmetic Groups, Invent. Math., Volume 70 (1982), pp. 169-218 | Article | MR 684171 | Zbl 0508.20020

[ZU2] S. Zucker Satake Compactifications, Comment. Math. Helv., Volume 58 (1983), pp. 312-343 | Article | MR 705539 | Zbl 0565.22009