A unit theorem for products of groups with several ends and Applications

TL;DR

This paper characterizes the unit groups of orders in group algebras for groups with multiple ends, providing a classification and applications related to torsion-free complements and simplified proofs of existing results.

Contribution

It offers a classification of unit groups in group algebras for groups with several ends, extending Kleinert's problem and providing new applications.

Findings

Classifies unit groups for groups virtually a product with multiple ends

Establishes existence of torsion-free normal complements

Provides simplified proofs of key results in algebra

Abstract

In his $1994$ survey, Kleinert defined formally and formulated the problem to obtain unit theorems for unit groups of orders in a semisimple algebra $A$. If $A$ is a group algebra $FG$, it boils down to classifying all finite groups $G$ such that the unit groups of most orders in $FG$ belong to a prescribed class $\mathcal{G}$ of infinite groups. We solve this problem for $\mathcal{G}$ consisting of the groups which are virtually a direct product of groups whose Cayley graph has more than one end. Subsequently, we obtain two types of applications. A first type being about the existence of torsion-free normal complements and a second about obtaining short and uniform proofs of some of the main results in [13,18,14,15].