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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
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    $\begingroup$ 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. $SL_n(\mathbb{Z})$. $\endgroup$ 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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    $\begingroup$ For reference, the result in the linked paper is Proposition 2.7. The argument is actually easy. Suppose there is a non-residually finite hyperbolic group. Then, by a couple of standard theorems, there is an infinite hyperbolic group $G$ with property (T) and no proper finite quotients. But it is residually amenable, and every amenable quotient of a (T) group is finite, contradiction. $\endgroup$
    – HJRW
    Jan 23 '21 at 12:06

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