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Formulas of Matsushima, of Garland and property (T) for groups acting on symmetric spaces or buildings. (Formules de Matsushima, de Garland et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles.) (French) Zbl 0933.22009
There is a whole bunch of so-called vanishing theorems, associated with the names of Matsushima and Garland. The Matsushima type theorems concern locally symmetric spaces, while Garland’s theorem relates to buildings. The author presents a very instructive survey of and geometric explanations for such results and gives new proofs for one variant of the Matsushima theorem and for Garland’s theorem. The main emphasis of the paper, however, is to show that these formulae yield Kazhdan’s property \((T)\). Thus let \(X\) be an irreducible symmetric space (distinct from the real or complex hyperbolic space) or a triangle building. Suppose that \(\Gamma\) is a discrete and cocompact group of isometries of \(X\), and let \(\rho\) be any unitary representation of \(\Gamma\). Then the first and second cohomology groups associated to \(\rho\) have the property that \(H^1(\Gamma,\rho)=0\) and \(H^2 (\Gamma, \rho)\) is separated. In particular, \(\Gamma\) has property \((T)\) since this is equivalent to that \(H^1(\Gamma,\rho)=0\) for all unitary representations \(\rho\) of \(\Gamma\).

MSC:
22D10 Unitary representations of locally compact groups
22E40 Discrete subgroups of Lie groups
51E24 Buildings and the geometry of diagrams
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[1] BOREL (A.) . - Cohomologie de certains groupes discrets et laplaciens p-adiques [d’après H. Garland] , Séminaire Bourbaki, exposé n^\circ 437, 1973 . Numdam | Zbl 0376.22009 · Zbl 0376.22009 · numdam:SB_1973-1974__16__12_0 · eudml:109845
[2] BARRE (S.) . - Polyèdres finis de dimension 2 à courbure \leq 0 et de rang 2 , Ann. Inst. Fourier, t. 45, 1995 , p. 1037-1059. Numdam | MR 96k:53056 | Zbl 0831.53031 · Zbl 0831.53031 · doi:10.5802/aif.1483 · numdam:AIF_1995__45_4_1037_0 · eudml:75143
[3] BALLMANN (W.) , BRIN (M.) . - Orbihedra of nonpositive curvature , Publ. Math., Institut Hautes Études Scientifiques, t. 82, 1995 , p. 169-209. Numdam | MR 97i:53049 | Zbl 0866.53029 · Zbl 0866.53029 · doi:10.1007/BF02698640 · numdam:PMIHES_1995__82__169_0 · eudml:104108
[4] BRIDSON (M.) , HAEFLIGER (A.) . - CAT(0) spaces , livre à paraître.
[5] BURGER (M.) , MOZES (S.) . - Finitely presented groups and products of trees , C. R. Acad. Sci. Paris, t. 324, 1997 , p. 747-752. MR 98g:20041 | Zbl 0966.20013 · Zbl 0966.20013 · doi:10.1016/S0764-4442(97)86938-8
[6] BOREL (A.) , SERRE (J.-P.) . - Corners and arithmetic groups , Comment. Math. Helvet., t. 48, 1973 , p. 436-491. MR 52 #8337 | Zbl 0274.22011 · Zbl 0274.22011 · doi:10.1007/BF02566134 · eudml:139559
[7] BALLMANN (W.) , SWIATKOWSKI (J.) . - On L2-cohomology and property (T) for automorphism groups of cell complexes . - Preprint, Bonn, 1996 .
[8] BURGER (M.) . - Kazhdan constants for SL (3, \Bbb Z) , J. Reine angew. Math., t. 385, 1991 , p. 203-220. MR 92c:22013
[9] BROWN (K.) . - Cohomology of groups , Graduate Texts in Mathematics, t. 87, Springer, Berlin, 1982 . MR 83k:20002 | Zbl 0584.20036 · Zbl 0584.20036
[10] BOREL (A.) , WALLACH (N.) . - Continuous cohomology, discrete subgroups and representations of reductive groups , Princeton University Press, Princeton, 1980 . MR 83c:22018 | Zbl 0443.22010 · Zbl 0443.22010
[11] CORLETTE (K.) . - Archimedean superrigidity and hyperbolic geometry , Ann. of Math., t. 135, 1992 , p. 195-182. MR 92m:57048 | Zbl 0768.53025 · Zbl 0768.53025 · doi:10.2307/2946567
[12] CARTWRIGHT (D.I.) , MLOTKOWSKI (W.) , STEGER (T.) . - Property (T) and \?2 groups , Ann. Inst. Fourier, t. 44, 1993 , p. 213-248. Numdam | MR 95j:20024 | Zbl 0792.43002 · Zbl 0792.43002 · doi:10.5802/aif.1395 · numdam:AIF_1994__44_1_213_0 · eudml:75056
[13] CARTWRIGHT (D.I.) , MANTERO (A.-M.) , STEGER (T.) , ZAPPA (A.) . - Groups acting simply transitively on the vertices of a building of type \?2 , Geom. Dedic., t. 47, I, p. 143-166, 1993 ; II, p. 167-223, 1993 . MR 95b:20053 | Zbl 0784.51011 · Zbl 0784.51011 · doi:10.1007/BF01266618
[14] DE LA HARPE (P.) , VALETTE (A.) . - La propriété (T) de Kazhdan pour les gropes localement compacts , Astérisque, Soc. Math. de France, t. 175, 1989 . · Zbl 0759.22001
[15] FEIT (W.) , HIGMAN (G.) . - The nonexistence of certain generalized polygons , J. of Algebra, t. 1, 1964 , p. 114-131. MR 30 #1189 | Zbl 0126.05303 · Zbl 0126.05303 · doi:10.1016/0021-8693(64)90028-6
[16] GARLAND (H.) . - p-adic curvature and the cohomoloty of discrete subgroups of p-adic groups , Ann. of Math., t. 97, 1973 , p. 375-423. MR 47 #8719 | Zbl 0262.22010 · Zbl 0262.22010 · doi:10.2307/1970829
[17] GARLAND (H.) , RAGHUNATHAN (M.S.) . - Fundamental domains for lattices in \Bbb R-rank 1 semisimple Lie groups , Ann. Math., t. 92, 1970 , p. 279-326. MR 42 #1943 | Zbl 0206.03603 · Zbl 0206.03603 · doi:10.2307/1970838
[18] GUICHARDET (A.) . - Cohomologie des groupes topologiques et des algèbres de Lie . - CEDIC-Fernand Nathan, Paris, 1980 . MR 83f:22004 | Zbl 0464.22001 · Zbl 0464.22001
[19] KOREVAAR (N.) , SCHOEN (R.) . - Sobolev spaces and harmonic maps for metric space targets , Commun. Anal. Geom., t. 1, 1993 , p. 561-659. MR 95b:58043 | Zbl 0862.58004 · Zbl 0862.58004
[20] KANIUTH (E.) , TAYLOR (K.F.) . - Kazhdan constants and the dual space topology , Math. ann., t. 293, 1992 , p. 495-508. MR 93i:22003 | Zbl 0787.22006 · Zbl 0787.22006 · doi:10.1007/BF01444731 · eudml:164972
[21] LOTT (J.) . - The zero in the spectrum conjecture . - Preprint, Univ. Michigan, 1996 . · Zbl 0874.58086
[22] LUBOTSKY (A.) . - Discrete grops, expanding graphs and invariant measures , Progress in Math., t; 125, Birkh\?user, Basel, 1994 .
[23] LUBOTSKY (A.) , ZIMMER (R.) . - Variants of Kazhdan’s property for subgroups of semi-simple groups , Israel J. Math., t. 66, 1989 , p. 289-299. Zbl 0706.22010 · Zbl 0706.22010 · doi:10.1007/BF02765899
[24] MATSUSHIMA (Y.) . - On the first Betti number of compact quotient spaces of higher dimensional symmetric spaces , Annals. of Math., t. 75, 1962 , p. 312-330. MR 28 #1629 | Zbl 0118.38303 · Zbl 0118.38303 · doi:10.2307/1970176
[25] MOK (N.) . - Harmonic forms with values in locally constant Hilbert bundles , in Conference in honor of Jean-Pierre Kahane, Orsay 1993 , J. Fourier Anal. Appl., special issue, 1995 , p. 433-454. MR 97c:58008 | Zbl 0891.58001 · Zbl 0891.58001
[26] MOK (N.) , SIU (Y.-T.) , YEUNG (S.-K.) . - Geometric superrigidity , Invent. Math., t. 113, 1993 , p. 57-83. MR 94h:53079 | Zbl 0808.53043 · Zbl 0808.53043 · doi:10.1007/BF01244302 · eudml:144122
[27] NIBLO (G.) , REEVES (L.) . - Groups acting on CAT(0) cube complexes , Geometry and Topology, t. 1, 1997 , p. 1-7. MR 98d:57005 | Zbl 0887.20016 · Zbl 0887.20016 · doi:10.2140/gt.1997.1.1 · emis:journals/UW/gt/GTVol1/paper1.abs.html · eudml:119370
[28] PANSU (P.) . - Sous-groupes discrets des groupes de Lie : rigidité, arithméticité , Séminaire Bourbaki n^\circ 778, Astérisque, t. 227, 1995 , p. 69-105. Numdam | MR 96a:22021 | Zbl 0835.22011 · Zbl 0835.22011 · numdam:SB_1993-1994__36__69_0 · eudml:110194
[29] RONAN (M.) . - A construction of buildings with no rank 3 residues of spherical types , in Buildings and the geometry of diagrams, CIME conference at Como, 1984 , Lect. Notes, t. 1181, Springer, Berlin, 1986 , p. 242-248. MR 87k:51015 | Zbl 0588.51015 · Zbl 0588.51015
[30] RONAN (M.) , TITS (J.) . - Building buildings , Math. Ann., t. 278, 1987 , p. 291-302. MR 89e:51005 | Zbl 0628.51001 · Zbl 0628.51001 · doi:10.1007/BF01458072 · eudml:164293
[31] SWIATKOWSKI (J.) . - Polygonal complexes of nonpositive curvature : from local symmetry to global one . - Preprint, Univ. Wroclaw, 1994 .
[32] TITS (J.) . - A local approach to buidings , in The geometric vein, Coxeter Festschrift, Springer, Berlin, 1981 , p. 519-647. MR 83k:51014 | Zbl 0496.51001 · Zbl 0496.51001
[33] TITS (J.) . - Spheres of radius 2 in triangle buildings , in Finite geometries, buildings and related topics, Pingree Park Conference, 1988 , Clarendon Press, Oxford, 1990 , p. 17-28. MR 91i:51009 | Zbl 0744.51013 · Zbl 0744.51013
[34] WANG (McK.) , ZILLER (W.) . - Symmetric spaces and strongly isotropy irreducible spaces , Math. Ann., t. 296, 1993 , p. 285-326. MR 94g:53042 | Zbl 0804.53075 · Zbl 0804.53075 · doi:10.1007/BF01445107 · eudml:165087
[35] ZUK (A.) . - La propriété (T) pour les groupes agissant sur les polyèdres , C.R. Acad. Sci. Paris, t. 323, 1996 , p. 453-458. MR 97i:22001 | Zbl 0858.22007 · Zbl 0858.22007
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