A unit theorem for products of groups with several ends and Applications
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917029250859008 |
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| author | Janssens, Geoffrey |
| author_facet | Janssens, Geoffrey |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18051 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A unit theorem for products of groups with several ends and Applications Janssens, Geoffrey Group Theory Rings and Algebras Representation Theory 20C05, 20E06, 16S34 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]. |
| title | A unit theorem for products of groups with several ends and Applications |
| topic | Group Theory Rings and Algebras Representation Theory 20C05, 20E06, 16S34 |
| url | https://arxiv.org/abs/2510.18051 |