Locally compact modules over abelian groups and compactly generated metabelian groups
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916921537986560 |
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| author | Cornulier, Yves |
| author_facet | Cornulier, Yves |
| contents | We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a devissage result for such modules of the form "compact-by-sheer-by-discrete", and then study more specifically the sheer part. The main typical example of a sheer module is a polycontractable module, i.e., a finite direct product of modules, each of which is contracted by some group element. We show that every sheer module has a "large" polycontractable submodule, in a suitable sense.
We apply this to the study of compactly generated metabelian groups. For instance, we prove that they always have a maximal compact normal subgroup, and we extend the Bieri-Strebel characterization of compactly presentable metabelian groups from the discrete case to this more general setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11360 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Locally compact modules over abelian groups and compactly generated metabelian groups Cornulier, Yves Group Theory Commutative Algebra Primary 13C05, Secondary 13E05, 22B05, 22D05 We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a devissage result for such modules of the form "compact-by-sheer-by-discrete", and then study more specifically the sheer part. The main typical example of a sheer module is a polycontractable module, i.e., a finite direct product of modules, each of which is contracted by some group element. We show that every sheer module has a "large" polycontractable submodule, in a suitable sense. We apply this to the study of compactly generated metabelian groups. For instance, we prove that they always have a maximal compact normal subgroup, and we extend the Bieri-Strebel characterization of compactly presentable metabelian groups from the discrete case to this more general setting. |
| title | Locally compact modules over abelian groups and compactly generated metabelian groups |
| topic | Group Theory Commutative Algebra Primary 13C05, Secondary 13E05, 22B05, 22D05 |
| url | https://arxiv.org/abs/2311.11360 |