Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866912209523703808 |
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| author | Castellano, Ilaria Marchionna, Bianca Weigel, Thomas |
| author_facet | Castellano, Ilaria Marchionna, Bianca Weigel, Thomas |
| contents | It is shown that a Stallings--Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Thm. B). More precisely, a compactly generated $\mathcal{CO}$-bounded t.d.l.c. group $G$ of rational discrete cohomological dimension less than or equal to $1$ must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody's rational version of the classical Stallings--Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension $1$ has necessarily non-positive Euler--Poincaré characteristic (cf. Thm. H). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_10847 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one Castellano, Ilaria Marchionna, Bianca Weigel, Thomas Group Theory 22D05, 20J05, 20J06 It is shown that a Stallings--Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Thm. B). More precisely, a compactly generated $\mathcal{CO}$-bounded t.d.l.c. group $G$ of rational discrete cohomological dimension less than or equal to $1$ must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody's rational version of the classical Stallings--Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension $1$ has necessarily non-positive Euler--Poincaré characteristic (cf. Thm. H). |
| title | Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one |
| topic | Group Theory 22D05, 20J05, 20J06 |
| url | https://arxiv.org/abs/2201.10847 |