# Are Hyperbolic Groups Residually Amenable

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?

• What are known examples of groups which are residually amenable but not residually finite? Jun 11 '13 at 14:41
• Since f.g. Nilpotent groups are residually finite, at least residually nilpotent always implies residually finite. Jun 11 '13 at 14:44
• @unknown(google): Baumslag solitar groups are residually solvable, hence residually amenable. Certain of these are not residually finite. Jun 11 '13 at 14:52
• @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. $SL_n(\mathbb{Z})$. Jun 11 '13 at 14:54

• 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.