Automorphism groups of Boolean powers with ample generics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908556744196096 |
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| author | Mayr, Peter Ruškuc, Nik |
| author_facet | Mayr, Peter Ruškuc, Nik |
| contents | Let $A$ be a finite non-abelian simple Mal'cev algebra, such as for example a finite simple non-abelian group or a finite simple non-zero ring. We show that the automorphism group of a filtered Boolean power of $A$ by the countable atomless Boolean algebra $A$ has ample generics. This uses the decomposition of that automorphism group as a semidirect product of a certain closure of a Boolean power of the automorphism group of $A$ by $B$ and the stabiliser of finitely many points in the homeomorphism group Homeo$2^ω$ of the Cantor space $2^ω$ by the authors. As an intermediate step, we show that pointwise stabilisers in Homeo$2^ω$ have ample generics, which extends the result of Kwiatkowska that Homeo$2^ω$ has ample generics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automorphism groups of Boolean powers with ample generics Mayr, Peter Ruškuc, Nik Rings and Algebras Group Theory Logic 20B27 (03E15, 06E15, 08A35, 22A05, 54H10) Let $A$ be a finite non-abelian simple Mal'cev algebra, such as for example a finite simple non-abelian group or a finite simple non-zero ring. We show that the automorphism group of a filtered Boolean power of $A$ by the countable atomless Boolean algebra $A$ has ample generics. This uses the decomposition of that automorphism group as a semidirect product of a certain closure of a Boolean power of the automorphism group of $A$ by $B$ and the stabiliser of finitely many points in the homeomorphism group Homeo$2^ω$ of the Cantor space $2^ω$ by the authors. As an intermediate step, we show that pointwise stabilisers in Homeo$2^ω$ have ample generics, which extends the result of Kwiatkowska that Homeo$2^ω$ has ample generics. |
| title | Automorphism groups of Boolean powers with ample generics |
| topic | Rings and Algebras Group Theory Logic 20B27 (03E15, 06E15, 08A35, 22A05, 54H10) |
| url | https://arxiv.org/abs/2509.20121 |