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Let $G$ be a compact topological group, and $\operatorname{Aut}(G)$ the group of autohomeomorphisms of $G$. I have proved some (topological) results about the holomorph $G\leftthreetimes \operatorname{Aut}(G)$, and now looking for nice examples.

I can, of course, take any compact group $G$, but I want to know if there are "natural" examples to be found. That is, are there any compact groups $G$ for which the semidirect product $G\leftthreetimes \operatorname{Aut}(G)$ is of a particular interest?

Thank you!

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    $\begingroup$ Is that really the notation people use for the topological semi-direct product? I've never seen this symbol used as an operation. $\endgroup$
    – David White
    Sep 19, 2015 at 18:55
  • $\begingroup$ Automorphisms, you mean? (if you only assume homeos, the semi-direct product makes no sense...) $\endgroup$
    – Alain Valette
    Sep 19, 2015 at 21:08
  • $\begingroup$ @DavidWhite : yes, this is the notation people use, as far as I know. See, for example, this recent survey u.math.biu.ac.il/~megereli/DMsurveyFinal.pdf $\endgroup$
    – Ludolila
    Sep 20, 2015 at 15:45
  • $\begingroup$ @AlainValette: I mean automorphisms that are also homeomorphisms. $\endgroup$
    – Ludolila
    Sep 20, 2015 at 15:46
  • $\begingroup$ OK, continuous automorphisms, then! You could look at cases where $Aut(G)$ is large, e.g. $G=C_2^\mathbb{N}$, where $C_2$ is the 2-element group. $\endgroup$
    – Alain Valette
    Sep 20, 2015 at 17:10

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