On the degree of rapid decay

Nica, Bogdan

doi:10.1090/S0002-9939-10-10289-5

On the degree of rapid decay
HTML articles powered by AMS MathViewer

by Bogdan Nica PDF

Proc. Amer. Math. Soc. 138 (2010), 2341-2347 Request permission

Abstract:

A finitely generated group $\Gamma$ equipped with a word-length is said to satisfy property RD if there are $C, s\geq 0$ such that, for all non-negative integers $n$, we have $\|a\|\leq C (1+n)^s \|a\|_2$ whenever $a\in \mathbb {C}\Gamma$ is supported on elements of length at most $n$.

We show that, for infinite $\Gamma$, the degree $s$ is at least $1/2$.

References

J. A. Behrstock, Y. N. Minsky: Centroids and the rapid decay property in mapping class groups, arXiv:0810.1969
I. Chatterji and K. Ruane, Some geometric groups with rapid decay, Geom. Funct. Anal. 15 (2005), no. 2, 311–339. MR 2153902, DOI 10.1007/s00039-005-0508-9
Joel M. Cohen, Operator norms on free groups, Boll. Un. Mat. Ital. B (6) 1 (1982), no. 3, 1055–1065 (English, with Italian summary). MR 683492
Alain Connes and Henri Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), no. 3, 345–388. MR 1066176, DOI 10.1016/0040-9383(90)90003-3
Pierre de la Harpe, Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 14, 771–774 (French, with English summary). MR 972078
Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
Cornelia Druţu and Mark Sapir, Relatively hyperbolic groups with rapid decay property, Int. Math. Res. Not. 19 (2005), 1181–1194. MR 2147058, DOI 10.1155/IMRN.2005.1181
Kenneth J. Dykema and Pierre de la Harpe, Some groups whose reduced $C^*$-algebras have stable rank one, J. Math. Pures Appl. (9) 78 (1999), no. 6, 591–608 (English, with English and French summaries). MR 1708667, DOI 10.1016/S0021-7824(99)00015-X
Uffe Haagerup, An example of a nonnuclear $C^{\ast }$-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, DOI 10.1007/BF01410082
Paul Jolissaint, Rapidly decreasing functions in reduced $C^*$-algebras of groups, Trans. Amer. Math. Soc. 317 (1990), no. 1, 167–196. MR 943303, DOI 10.1090/S0002-9947-1990-0943303-2
Vincent Lafforgue, A proof of property (RD) for cocompact lattices of $\textrm {SL}(3,\mathbf R)$ and $\textrm {SL}(3,\mathbf C)$, J. Lie Theory 10 (2000), no. 2, 255–267. MR 1774859
Vincent Lafforgue, $K$-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math. 149 (2002), no. 1, 1–95 (French). MR 1914617, DOI 10.1007/s002220200213
Igor Mineyev and Guoliang Yu, The Baum-Connes conjecture for hyperbolic groups, Invent. Math. 149 (2002), no. 1, 97–122. MR 1914618, DOI 10.1007/s002220200214
M. Rajagopalan, On the $L^{p}$-space of a locally compact group, Colloq. Math. 10 (1963), 49–52. MR 148792, DOI 10.4064/cm-10-1-49-52
J. Ramagge, G. Robertson, and T. Steger, A Haagerup inequality for $\widetilde A_1\times \widetilde A_1$ and $\widetilde A_2$ buildings, Geom. Funct. Anal. 8 (1998), no. 4, 702–731. MR 1633983, DOI 10.1007/s000390050071
Sadahiro Saeki, The $L^p$-conjecture and Young’s inequality, Illinois J. Math. 34 (1990), no. 3, 614–627. MR 1053566