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Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, or induction from subgroups? Also, except the ones created by "general theory" of groups, like the amenable representation of a non-amenable non-property (T) group constructed by Bekka-Valette, etc. I'd like to see "concrete" particular representations for particular families of monster groups, representations that appear because of their very own special properties and structure.

This is a vague question. If you have a group in your mind, that is monstrous enough (mainly in the sense of being non-amenable but with amenable subgroups) and has an interesting representation, feel free to present it.

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    $\begingroup$ TM can be constructed as a limit of hyperbolic groups. This means you can take boundary unitary representations of the latter (Bader-Muchnik) and then try to construct some interesting limit. Doing so would be quite interesting; I do not think anybody is working on unitary theory of lacunary hyperbolic groups. See arxiv.org/pdf/math/0701365.pdf for the definitions. $\endgroup$
    – Misha
    Nov 21, 2014 at 18:43
  • $\begingroup$ There is no universal definition of Tarski monster, could you give one? Do you mean an infinite f.g. quasi-finite group (quasi-finite = in which every proper subgroup is finite)? or something else such as "every proper subgroup is cyclic"? $\endgroup$
    – YCor
    Nov 23, 2014 at 14:36

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