Simplicial cohomology of augmentation ideals in ${\ell}^1(G)$

TL;DR

This paper investigates the Hochschild cohomology of the group algebra (G) for discrete groups, establishing a decomposition theorem and linking bounded cohomology with simplicial cohomology for certain groups.

Contribution

It introduces a decomposition theorem for Hochschild cohomology of (G) and shows an isomorphism between bounded and simplicial cohomology for commutative-transitive groups.

Findings

Decomposition theorem for Hochschild cohomology of (G)

Isomorphism between bounded and simplicial cohomology in specific groups

Extension of cohomological understanding in group algebras

Abstract

Let $G$ be a discrete group. We give a decomposition theorem for the Hochschild cohomology of $\ell^1(G)$ with coefficients in certain $G$-modules. Using this we show that if $G$ is commutative-transitive, the canonical inclusion of bounded cohomology of $G$ into simplicial cohomology of $\ell^1(G)$ is an isomorphism.