×

Transition operators on co-compact \(G\)-spaces. (English) Zbl 1116.22007

The authors prove the following result: Let \(M\) be a complete, co-compact Riemannian manifold. Then \(0\) is in the \(L^2\)-spectrum of the Laplacian of \(M\) if and only if some (equivalently, every) closed quasi-transitive group of isometries of \(M\) is both amenable and unimodular. The same result holds also for Euclidean simplicial complexes.

MSC:

22F50 Groups as automorphisms of other structures
43A07 Means on groups, semigroups, etc.; amenable groups
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
58C40 Spectral theory; eigenvalue problems on manifolds
58J05 Elliptic equations on manifolds, general theory
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

References:

[1] Berg, C. and Christensen, J. P. R.: On the relation between amenability of locally compact groups and the norms of convolution operators. Math. Ann. 208 (1974), 149-153. · Zbl 0264.43003 · doi:10.1007/BF01432382
[2] Berg, C. and Christensen, J. P. R.: Sur la norme des opérateurs de convolution. Invent. Math. 23 (1974), 173-178. · Zbl 0261.22009 · doi:10.1007/BF01405169
[3] Bourbaki, N.: Intégration . Hermann, Paris 1963.
[4] Brin, M. and Kifer, Y.: Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes. Math. Z. 237 (2001), 421-468. · Zbl 0984.58022 · doi:10.1007/PL00004875
[5] Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv. 56 (1981), 581-598. · Zbl 0495.58029 · doi:10.1007/BF02566228
[6] Cartwright, D. I., Kaimanovich, V. A. and Woess, W.: Random walks on the affine group of local fields and of homogeneous tree. Ann. Inst. Fourier (Grenoble) 44 (1994), 1243-1288. · Zbl 0809.60010 · doi:10.5802/aif.1433
[7] Cattaneo, C.: The spectrum of the continuous Laplacian on a graph. Monatsh. Math. 124 (1997), 215-235. · Zbl 0892.47001 · doi:10.1007/BF01298245
[8] Chatterji, I. L., Pittet, Ch. and Saloff-Coste, L.: Connected Lie groups and property RD. To appear in Duke Math. J. . · Zbl 1119.22006 · doi:10.1215/S0012-7094-07-13733-5
[9] Copson, E. T.: Metric Spaces . Cambridge Tracts in Mathematics and Mathematical Physics 57 . Cambridge University Press, London, 1968. · Zbl 0177.25303
[10] Cowling, M.: Michael Herz’s “principe de majoration” and the Kunze-Stein phenomenon. In Harmonic analysis and number theory (Montreal, 1996) , 73-88. CMS Conf. Proc. 21 . Amer. Math. Soc., Providence, RI, 1997. · Zbl 0964.22008
[11] Davies, E. B.: Non-Gaussian aspects of heat kernel behaviour. J. London Math. Soc. (2) 55 (1997), 105-125. · Zbl 0879.35064 · doi:10.1112/S0024610796004607
[12] Day, M. M.: Convolutions, means and spectra. Illinois J. Math. 8 (1964), 100-111. · Zbl 0122.11703
[13] Dellacherie, C. and Meyer, P.-A.: Probabilités et Potentiel, Chapitres I à IV . Actualités Scientifiques et Industrielles 1372 . Hermann, Paris, 1975. · Zbl 0323.60039
[14] Derriennic, Y. and Guivarc’h, Y.: Théorème de renouvellement pour les groupes non moyennables. C. R. Acad. Sci. Paris, Sér. A-B 277 (1973), A613-A615. · Zbl 0272.60005
[15] Dieck, T. tom: Transformation Groups . De Gruyter Studies in Mathematics 8 . Walter de Gruyter, Berlin, 1987. · Zbl 0611.57002
[16] Eells, J. and Fuglede, B.: Harmonic maps between Riemannian polyhedra . Cambridge Tracts in Mathematics 142 . Cambridge University Press, Cambridge, 2001. · Zbl 0979.31001
[17] Farb, B. and Mosher, L.: A rigidity theorem for the solvable Baumslag-Solitar groups. Invent. Math. 131 (1998), 419-451. · Zbl 0937.22003 · doi:10.1007/s002220050210
[18] Guivarc’h, Y.: Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 333-379. · Zbl 0294.43003
[19] Guivarc’h, Y.: Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. In Conference on Random Walks (Kleebach, 1979) , 47-98. Astérisque 74 . Soc. Math. France, Paris, 1980. · Zbl 0448.60007
[20] Herz, C.: Sur le phénomène de Kunze-Stein. C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A491-A493. · Zbl 0198.18202
[21] Herz, C.: The theory of \(p\)-spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154 (1971), 69-82. JSTOR: · Zbl 0216.15606 · doi:10.2307/1995427
[22] Hewitt, E. and Ross, K. A.: Abstract Harmonic Analysis Vol. I: Structure of topological groups. Integration theory, group representations . Die Grundlehren der mathematischen Wissenschaften 115 . Academic Press, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. · Zbl 0115.10603
[23] Kesten, H.: Full Banach mean values on countable groups. Math. Scand. 7 (1959), 146-156. · Zbl 0092.26704
[24] Kobayashi, T., Ono, K. and Sunada, T.: Periodic Schrödinger operators on a manifold. Forum Math. 1 (1989), 69-79. · Zbl 0655.58033 · doi:10.1515/form.1989.1.69
[25] Lohoué, N.: Estimations \(L^p\) des coefficients de représentation et opérateurs de convolution. Adv. in Math. 38 (1980), 178-221. · Zbl 0463.43003 · doi:10.1016/0001-8708(80)90004-3
[26] Nummelin, E.: General irreducible Markov chains and nonnegative operators . Cambridge Tracts in Mathematics 83 . Cambridge University Press, Cambridge, 1984. · Zbl 0551.60066 · doi:10.1017/CBO9780511526237
[27] Parthasarathy, K. R.: Probability measures on metric spaces . Probability and Mathematical Statistics 3 . Academic Press, New York, 1967. · Zbl 0153.19101
[28] Pittet, Ch.: The isoperimetric profile of homogeneous Riemannian manifolds. J. Differential Geom. 54 (2000), 255-302. · Zbl 1035.53069
[29] Raghunathan, M. S.: Discrete subgroups of Lie groups . Ergebnisse der Mathematik und ihrer Grenzgebiete 68 . Springer-Verlag, New York-Heidelberg, 1972. · Zbl 0254.22005
[30] Reiter, H. and Stegeman, J. D.: Classical harmonic analysis and locally compact groups . London Mathematical Society Monographs New Series 22 . The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0965.43001
[31] Rudin, W.: Real and complex analysis . McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966. · Zbl 0142.01701
[32] Saloff-Coste, L.: Aspects of Sobolev-type inequalities . London Math. Soc. Lecture Note Series 289 . Cambridge Univ. Press, Cambridge, 2002. · Zbl 0991.35002
[33] Saloff-Coste, L. and Woess, W.: Computing norms of group-invariant transition operators. Combin. Probab. Comput. 5 (1996), 161-178. · Zbl 0865.60006 · doi:10.1017/S0963548300001942
[34] Saloff-Coste, L. and Woess, W.: Transition operators, groups, norms, and spectral radii. Pacific J. Math. 180 (1997), 333-367. · Zbl 0899.60005 · doi:10.2140/pjm.1997.180.333
[35] Salvatori, M.: On the norms of group-invariant transition operators on graphs. J. Theoret. Probab. 5 (1992), 563-576. · Zbl 0751.60068 · doi:10.1007/BF01060436
[36] Schaefer, H. H.: Banach Lattices and Positive Operators . Die Grundlehren der mathematischen Wissenschaften 215 . Springer-Verlag, New York-Heidelberg, 1974. · Zbl 0296.47023
[37] Seneta, E.: Non-negative matrices and Markov chains . Second edition. Springer Series in Statistics. Springer-Verlag, New York, 1981. · Zbl 0471.60001
[38] Soardi, P. M. and Woess, W.: Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z. 205 (1990), 471-486. · Zbl 0693.43001 · doi:10.1007/BF02571256
[39] Sturm, K.-Th.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 (1995), 275-312. · Zbl 0854.35015
[40] Sy, P. W. and Sunada, T.: Discrete Schrödinger operators on a graph. Nagoya Math. J. 125 (1992), 141-150. · Zbl 0773.35046
[41] Varopoulos, N. Th.: Small time Gaussian estimates of heat diffusion kernels. I: The semigroup technique. Bull. Sci. Math. 113 (1989), 253-277. · Zbl 0703.58052
[42] Woess, W.: Périodicité de mesures de probabilité sur les groupes topologiques. In Random walks and stochastic processes on Lie groups (Nancy, 1981) , 170-180. Inst. Élie Cartan 7 . Univ. Nancy, Nancy, 1983. · Zbl 0495.60016
[43] Żuk, A.: A generalized Følner condition and the norms of random walk operators on groups. Enseig. Math. (2) 45 (1999), 321-348. · Zbl 0990.43001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.