Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture

Details Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture

2007-07-25

arxiv.org/abs/0707.3658v3

J. Noncommut. Geom. 4 (2010), 83-124

Mathematics

K-Theory and Homology

32 pages, 2 figures; added an appendix also by C. Ogle

Scientific paper

10.4171/JNCG/50

By deploying dense subalgebras of $\ell^1(G)$ we generalize the Bass conjecture in terms of Connes' cyclic homology theory. In particular, we propose a stronger version of the $\ell^1$-Bass Conjecture. We prove that hyperbolic groups relative to finitely many subgroups, each of which posses the polynomial conjugacy-bound property and nilpotent periodicity property, satisfy the $\ell^1$-Stronger-Bass Conjecture. Moreover, we determine the conjugacy-bound for relatively hyperbolic groups and compute the cyclic cohomology of the $\ell^1$-algebra of any discrete group.

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