Local cyclic homology of group Banach algebras I: Hyperbolic groups

TL;DR

This paper computes the local cyclic cohomology of Banach algebras associated with hyperbolic groups, showing the algebraic assembly map is an isomorphism, which advances understanding in noncommutative geometry.

Contribution

It provides the first explicit calculation of local cyclic cohomology for Banach algebras of hyperbolic groups, establishing the isomorphism of the assembly map in this context.

Findings

Local cyclic cohomology computed for hyperbolic groups

Assembly map in local cyclic homology is an isomorphism for these groups

Advances understanding of noncommutative invariants of hyperbolic groups

Abstract

We calculate the bivariant local cyclic cohomology of the Banach convolution algebra of summable functions on a word-hyperbolic group. Our result implies that the Banach algebraic assembly map in local cyclic homology is an isomorphism for such a group.