Is there a group $G$ with the following properties?

  1. Finitely presented
  2. Amenable
  3. Not virtually solvable
  4. LERF (that is, every finitely generated subgroup is closed in the profinite topology on $G$).

If this is hard, I can relax requirement 4: Instead of LERF, it's enough that every finitely generated subgroup which has finitely many conjugates (that is, normalizer of finite index in $G$) is closed in the profinite topology on $G$.

  • $\begingroup$ As far as I know, Grigorchuk has an example satisfying 1., 2., 3., but not 4. dx.doi.org/10.1070/SM1998v189n01ABEH000293 But this group is not residually finite, much less LERF. $\endgroup$ – Ian Agol Apr 22 '16 at 16:07
  • $\begingroup$ Probably this is an open question (which naturally splits into two cases, the elementary amenable and the non-elementary-amenable case). @IanAgol when 4 is relaxed there are also elementary amenable examples, such as Houghton groups discovered in the 70's. $\endgroup$ – YCor Apr 24 '16 at 22:40

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