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

TL;DR

This paper extends the Bass conjecture using cyclic homology, proving it for certain relatively hyperbolic groups and computing related algebraic invariants, thus advancing understanding of group algebras and their properties.

Contribution

It generalizes the Bass conjecture via cyclic homology and proves it for a class of relatively hyperbolic groups with specific properties.

Findings

Proves the $ ext{l}^1$-Stronger-Bass Conjecture for certain relatively hyperbolic groups.

Determines the conjugacy-bound for these groups.

Computes the cyclic cohomology of the $ ext{l}^1$-algebra of any discrete group.

Abstract

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.