# Finitely presented amenable LERF group which is not virtually solvable

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 $G$.

• 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. – Ian Agol Apr 22 '16 at 16:07
• 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. – YCor Apr 24 '16 at 22:40