Geodesic
Spherical Mean and the Fundamental Group
Canadian mathematical bulletin, Tome 34 (1991) no. 1, pp. 3-11

Voir la notice de l'article provenant de la source Cambridge University Press

We investigate some properties of spherical means on the universal covering space of a compact Riemannian manifold. If the fundamental group is amenable then the greatest lower bounds of the spectrum of spherical Laplacians are equal to zero. If the fundamental group is nontransient so are geodesic random walks. We also give an isoperimetric inequality for spherical means.
DOI : 10.4153/CMB-1991-001-2
Mots-clés : Spherical mean, geodesic random walk, amenable, transient, isoperimetric inequality, 58C40, 60J15 
Adachi, Toshiaki. Spherical Mean and the Fundamental Group. Canadian mathematical bulletin, Tome 34 (1991) no. 1, pp. 3-11. doi: 10.4153/CMB-1991-001-2
@article{10_4153_CMB_1991_001_2,
     author = {Adachi, Toshiaki},
     title = {Spherical {Mean} and the {Fundamental} {Group}},
     journal = {Canadian mathematical bulletin},
     pages = {3--11},
     year = {1991},
     volume = {34},
     number = {1},
     doi = {10.4153/CMB-1991-001-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1991-001-2/}
}
TY  - JOUR
AU  - Adachi, Toshiaki
TI  - Spherical Mean and the Fundamental Group
JO  - Canadian mathematical bulletin
PY  - 1991
SP  - 3
EP  - 11
VL  - 34
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1991-001-2/
DO  - 10.4153/CMB-1991-001-2
ID  - 10_4153_CMB_1991_001_2
ER  - 
%0 Journal Article
%A Adachi, Toshiaki
%T Spherical Mean and the Fundamental Group
%J Canadian mathematical bulletin
%D 1991
%P 3-11
%V 34
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1991-001-2/
%R 10.4153/CMB-1991-001-2
%F 10_4153_CMB_1991_001_2

[1] 1. Besse, A. L., Manifolds all of whose geodesies are closed. Springer-Verlag, 1978. Google Scholar

[2] 2. Brooks, R., The fundamental group and the spectrum of the Laplacian, Comm. Math. Helv. 56(1981), 581– 598. Google Scholar

[3] 3. Brooks, R., A relation between growth and the spectrum of the Laplacian, Math. Z. 198 (1981), 501–508. Google Scholar

[4] 4. Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, Princeton Univ. Press (1970), 195–199. Google Scholar

[5] 5. Dodziuk, J., Difference equations, isoperimetric inequality and transience of certain random walks, Trans. A.M.S. 284 (1984), 787–794. Google Scholar

[6] 6. Følner, E., On groups with full Banach mean value, Math. Scand. 3 (1955), 243–254. Google Scholar

[7] 7. Fukushima, M., Dirichlet forms and Markov processes. North-Holland, 1980. Google Scholar

[8] 8. Guivarc'h, Y., Keane, M. and Roynette, B., Marches Aléatoires sur les groupes de Lie. Springer Lecture Note in Math. 624, 1977. Google Scholar

[9] 9. Oshima, Y., Lecture on Dirichlet spaces, preprint. Google Scholar

[10] 10. Sunada, T., Spherical mean and geodesic chains on a Riemannian manifold, Trans. A.M.S. 267 (1981), 483–501. 11 , Geodesic flows and geodesic random walks, Advanced Studies in Pure Math. 3 (1984), 47–85. Google Scholar

[12] 12 Sunada, T., Unitary representations of fundamental groups and the spectrum of twisted Laplacian, preprint. Google Scholar

[13] 13. Varopoulos, N. Th., Brownian motion and transient groups, Ann. Inst. Fourier 33 (1983), 241–261. Google Scholar

Cité par Sources :