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

TL;DR

The paper advances unit theory for orders in finite-dimensional semisimple algebras by solving the Virtual Structure Problem for the broad class $\mathcal{G}_{\infty}$, i.e., products of groups with infinitely many ends. It reduces the problem to the Wedderburn–Artin decomposition of $FG$, identifies exact end-type constraints on unit groups $\mathcal{U}(RG)$, and derives a complete end-count classification (infinite ends, two ends, or zero ends) tied to the simple components of $FG$. These results yield uniform, ring-theoretic constructions of normal complements and Zassenhaus-type conclusions at the block level, applicable without exhaustive case-by-case group-by-group analysis. Overall, the work connects geometric group-theoretic notions of ends with algebraic unit theory to produce broad, practical unit-theorem consequences for orders in group algebras and their normal-structure properties.

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].

Theorems & Definitions (17)

• Theorem 1.1
• Corollary 1.2
• Remark
• Theorem 1.3
• Proposition 1.4
• Remark
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• proof