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- W4375959451 abstract "Let $G= Nrtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup $N$ and a closed subgroup $H.$ The Bohr compactification ${rm Bohr}(G)$ and the profinite completion ${rm Prof}(G)$ of $G$ are, respectively, isomorphic to semi-direct products $ Q_1 rtimes {rm Bohr}(H)$ and $ Q_2 rtimes {rm Prof}(H)$ for appropriate quotients $Q_1$ of ${rm Bohr}(N)$ and $Q_2$ of ${rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of $H$ on appropriate subsets of the dual space of $N$. In the case where $N$ is abelian, we have ${rm Bohr}(G)cong A rtimes {rm Bohr}(H)$ and ${rm Prof}(G)cong B rtimes {rm Prof}(H),$ where $A$ is the group of unitary characters of $N$ with finite $H$-orbits and $B$ the subgroup of $A$ of characters with finite image. Necessary and sufficient conditions are deduced for $G$ to be maximally almost periodic or residually finite. We apply the results to the case where $G= Lambdawr H$ is a wreath product of countable groups; we show in particular that ${rm Bohr}(Lambdawr H)$ is isomorphic to ${rm Bohr}(Lambda^{rm Ab}wr H)$ and ${rm Prof}(Lambdawr H)$ is isomorphic to ${rm Prof}(Lambda^{rm Ab} wr H),$ where $Lambda^{rm Ab}=Lambda/ [Lambda, Lambda]$ is the abelianization of $Lambda.$ As examples, we compute ${rm Bohr}(G)$ and ${rm Prof}(G)$ when $G$ is a lamplighter group and when $G$ is the Heisenberg group over a unital commutative ring." @default.
- W4375959451 created "2023-05-10" @default.
- W4375959451 creator A5003113893 @default.
- W4375959451 date "2023-05-08" @default.
- W4375959451 modified "2023-09-26" @default.
- W4375959451 title "On Bohr compactifications and profinite completions of group extensions" @default.
- W4375959451 doi "https://doi.org/10.48550/arxiv.2305.04803" @default.
- W4375959451 hasPublicationYear "2023" @default.
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