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?

  • $\begingroup$ 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
  • 2
    $\begingroup$ @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
  • 1
    $\begingroup$ @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

Proposition 7 of this paper establishes that every hyperbolic group is residually amenable iff every hyperbolic group is residually finite.

| cite | improve this answer | |
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.