Semidirect Product Compactifications

Dangello, Frank; Lindahl, R. J.

doi:10.4153/cjm-1983-001-7

Cited by 2 publications

(2 citation statements)

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“… The Bohr compactification of an abelian locally compact group A is easy to describe: can be identified with where is viewed as discrete group; in case A is finitely generated, a more precise description of is available (see [ Bek23 , proposition 11]). Provided and are known, Corollary F together with Corollary D give, in view of (i), a complete description of the Bohr compactification and the profinite completion of any wreath product in case X is infinite. Bohr compactifications of group and semigroup extensions have been studied by several authors, in a more abstract and less explicit setting ([ DL83 , JL81 , Jun78 , JM02 , Lan72 , Mil83 ]); profinite completions of group extensions appear at numerous places in the literature ([ GZ11 , RZ00 ]). …”

Section: Introductionmentioning

confidence: 99%

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On Bohr compactifications and profinite completions of group extensions

BEKKA

2023

Math. Proc. Camb. Phil. Soc.

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Let $G= N\rtimes H$ 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 (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively 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 $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation 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.

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Section: Introductionmentioning

confidence: 99%

“…Bohr compactifications of group and semigroup extensions have been studied by several authors, in a more abstract and less explicit setting ([ DL83 , JL81 , Jun78 , JM02 , Lan72 , Mil83 ]); profinite completions of group extensions appear at numerous places in the literature ([ GZ11 , RZ00 ]).…”

Section: Introductionmentioning

confidence: 99%