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I've encountered in Bass, Lubotzky, Magid, and Mozes - The proalgebraic completion of rigid groups (Remark 1. p. 7) the following terminology:

  • A normal subgroup $N$ of $G$ is observable if every $N$-representation can be embedded $N$-equivariantly in the restriction of a $G$-representation.

Unfortunately, when I've tried to google what is known about groups with this property the literature only seems to talk about algebraic groups. Is there another term which is normally used for this property in the context of discrete groups?

A second question: are there any immediately known classes of subgroups which are always observable?

I guess finite index subgroups have this property, but I don't know any others.

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  • $\begingroup$ How important is the hypothesis that the subgroup is normal? Retracts also have this property, but normal retracts are just direct factors, so that doesn't make a terribly interesting class of subgroups. $\endgroup$
    – HJRW
    Commented Mar 3, 2021 at 14:06
  • $\begingroup$ Does "representation" mean "f.-d. representation"? Otherwise it seems that you can always embed $V_N$ in the induced representation $\{f : G \to V_N \mathrel: f(h g) = h f(g)\}$ by sending $v$ to the extension by $0$ of $h \mapsto h v$. $\endgroup$
    – LSpice
    Commented Mar 3, 2021 at 14:22
  • $\begingroup$ @LSpice I agree that it should be finite dimensional representations. For what I want to do with them they should be normal subgroups, but I really just want to know if the property of being an observable subgroup goes by another name in the literature. $\endgroup$
    – Patrick Elliott
    Commented Mar 4, 2021 at 3:58
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    $\begingroup$ I very much doubt there is a standard terminology for this notion. $\endgroup$
    – Moishe Kohan
    Commented Mar 4, 2021 at 14:46

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