Abstract. By deploying dense subalgebras of`1.G/ we generalize the Bass conjecture in terms of Connes' cyclic homology theory. In particular, we propose a stronger version of thè 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`1-Stronger-Bass Conjecture. Moreover, we determine the conjugacy bound for relatively hyperbolic groups and compute the cyclic cohomology of the`1-algebra of any discrete group. (2010). 46L80, 20F65, 16S34. Mathematics Subject Classification

We establish the homological foundations for studying polynomially bounded group cohomology, and show that the natural map from P H * (G; Q) to H * (G; Q) is an isomorphism for a certain class of groups.

A discrete group with word-length (G, L) is B-isocohomological for a bounding classes B if the comparison map from B-bounded cohomology to ordinary cohomology (with coefficients in C) is an isomorphism; it is strongly B-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong B-isocohomologicality. In particular, we show strong B-isocohomologicality for an F P ∞ group G if all of the weighted G-sensitive Dehn functions are B-bounded. Such groups include all B-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of B-bounded cohomology of groups with respect to subgroups is introduced. Relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and a relative F P ∞ property for groups with respect to a collection of subgroups. Applications for computing B-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.

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