Nuclear Group Algebras for Finitely Generated Groups

TL;DR

This paper investigates the nuclearity of completions of group algebras for finitely generated groups, linking it to group growth properties and extending previous results to more general weights.

Contribution

It introduces new group algebras with broader weight functions and analyzes their nuclearity, expanding on Jolissaint's work and comparing different algebraic approaches.

Findings

Nuclearity is characterized by group growth properties.

Extended the class of weights beyond polynomial decrease.

Connected algebraic properties to the convergence of growth functions.

Abstract

We study completions of the group algebra of a finitely generated group and relate nuclearity of such a completion to growth properties of the group. This extends previous work of Jolissaint on nuclearity of rapidly decreasing functions on a finitely generated group to more general weights than polynomial decrease. The new group algebras and their duals are studied in detail and compared to other approaches. As application we discuss the convergence of the complete growth function introduced by Grigorchuk and Nagnibeda.