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It is a well-known conjecture (or maybe just a question) that all hyperbolic groups are residually finite. What happens if we weaken the conclusion; in particular

Are all hyperbolic groups residually amenable?

What is known in this direction?

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  • $\begingroup$ What are known examples of groups which are residually amenable but not residually finite? $\endgroup$ – ThiKu Jun 11 '13 at 14:41
  • $\begingroup$ Since f.g. Nilpotent groups are residually finite, at least residually nilpotent always implies residually finite. $\endgroup$ – ThiKu Jun 11 '13 at 14:44
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    $\begingroup$ @unknown(google): Baumslag solitar groups are residually solvable, hence residually amenable. Certain of these are not residually finite. $\endgroup$ – Jon Bannon Jun 11 '13 at 14:52
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    $\begingroup$ @unknown: In the world of discrete groups;There are amenable groups that are not residually finite, take for example the wreath product $G\wr H$ with $G$ and $H$ amenable and $G$ non-abelian, (in fact there are also infinite and simple ones). As far as non-amenable examples, just take one of the examples above and then take a (direct or free) product with a residually finite non-amenable group (eg. $SL_n(\mathbb{Z})$. $\endgroup$ – Owen Sizemore Jun 11 '13 at 14:54
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Proposition 7 of this paper establishes that every hyperbolic group is residually amenable iff every hyperbolic group is residually finite.

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  • $\begingroup$ You're welcome! The conjecture below Proposition 7 looks interesting, doesn't it? $\endgroup$ – Jon Bannon Jun 11 '13 at 15:10
  • $\begingroup$ In case the link goes down, the paper is: Asymptotic approximations of finitely generated groups, Goulnara Arzhantseva. $\endgroup$ – user35370 Apr 29 '15 at 18:28

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