What finitely generated amenable groups are known to be LERF?

Question

I know that finitely generated nilpotent groups are LERF (LERF means "subgroup separated").

I'm looking for examples (many, if possible) of groups which are:

1. Finitely generated, but infinite
2. Amenable
3. LERF
4. Not nilpotent

Answer 1

In "Growth of finitely generated solvable groups", by John Milnor, it is proved that if a finitely generated group $G$ is solvable group and is not polycyclic, then $G$ has exponential growth. Note that:

• Solvable groups are amenable.
• Finitely generated nilpotent groups are polycyclic. Hence a solvable group that is not polycyclic is not f.g. nilpotent. Alternatively you could use the fact that nilpotent groups have polynomial growth.
• A polycyclic group is a solvable group where every subgroup is finitely generated. Conveniently, grouprops has a page "Finitely generated and solvable not implies polycyclic" which gives a few examples, of f.g. solvable groups that are not polycyclic and a general strategy to constructing some.(Restricted) Wreath products provides a way to construct such groups, if $S \neq 1$ is solvable, then $S \wr \mathbb Z$ (the standard wreath product) is also solvable but it has a subgroup which is not finitely generated (the infinite direct sum of $S$).

Prop 3.19 in "Finitely presented wreath products and double coset decompositions", Yves de Cornulier proves that the wreath product $A \wr \mathbb Z$ is LERF, where $A$ is finitely generated abelian. Putting together this theorem and the above points we get that $A \wr \mathbb Z$ satisfies your conditions, when $A$ is a f.g. abelian group. For example the classic Lamplighter group, $\mathbb Z_2 \wr \mathbb Z$, works.

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I could not access the paper, but apparently it was proven in On homomorphisms onto finite groups by A.I. Mal'cev, that virtually polycyclic groups are LERF. This is mentioned in Yves de Cornulier paper "Finitely presented wreath products and double coset decompositions" Example 3.10.

Polycyclic groups don't have to be nilpotent. For example, in "Uniform Exponential Growth of Polycyclic Groups" by Roger C. Alperin, Thm 3.1 a condition is given for f.g. free abelian by cyclic groups to have exponential growth. These groups are polycyclic, by construction, but can't be (virtually) nilpotent since they don't have polynomial growth.

As another class of examples, if you extend a finite, but not polycyclic(solvable) group by an infinite polycyclic group you get more examples, since these are virtually polycyclic, but not polycyclic(solvable). These could have polynomial growth, since you could extend a f.g. nilpotent group(which are polycyclic).

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Grigorchuk group satisfies all you conditions and is not elementary amenable.