Projective simplicity and Bergman's property for unit groups of continuous rings
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909942643949568 |
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| author | Schneider, Friedrich Martin |
| author_facet | Schneider, Friedrich Martin |
| contents | We prove that the projective unit group $\mathrm{PGL}(R)$, i.e., the quotient of the unit group $\mathrm{GL}(R)$ modulo its center, of any non-discrete irreducible, continuous ring $R$ is simple. Moreover, we show that $\mathrm{GL}(R)$ has uncountable strong cofinality, that is, it is not the union of a countable chain of proper subgroups and it has finite width with respect to any generating set. Equivalently, every isometric action of $\mathrm{GL}(R)$ on a metric space has bounded orbits. It follows that every action of $\mathrm{GL}(R)$ by isometries on a non-empty complete $\mathrm{CAT}(0)$ space admits a fixed point. In particular, $\mathrm{GL}(R)$ possesses Serre's properties $(FH)$ and $(FA)$. Furthermore, our results entail that $\mathrm{PGL}(R)$ has bounded normal generation. In turn, we answer two questions by Carderi and Thom. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projective simplicity and Bergman's property for unit groups of continuous rings Schneider, Friedrich Martin Group Theory Rings and Algebras 16E50, 20E32, 20E45, 20F65 We prove that the projective unit group $\mathrm{PGL}(R)$, i.e., the quotient of the unit group $\mathrm{GL}(R)$ modulo its center, of any non-discrete irreducible, continuous ring $R$ is simple. Moreover, we show that $\mathrm{GL}(R)$ has uncountable strong cofinality, that is, it is not the union of a countable chain of proper subgroups and it has finite width with respect to any generating set. Equivalently, every isometric action of $\mathrm{GL}(R)$ on a metric space has bounded orbits. It follows that every action of $\mathrm{GL}(R)$ by isometries on a non-empty complete $\mathrm{CAT}(0)$ space admits a fixed point. In particular, $\mathrm{GL}(R)$ possesses Serre's properties $(FH)$ and $(FA)$. Furthermore, our results entail that $\mathrm{PGL}(R)$ has bounded normal generation. In turn, we answer two questions by Carderi and Thom. |
| title | Projective simplicity and Bergman's property for unit groups of continuous rings |
| topic | Group Theory Rings and Algebras 16E50, 20E32, 20E45, 20F65 |
| url | https://arxiv.org/abs/2510.23752 |