# Quasi-isometry classes of elementary amenable groups

Is there any elementary argument showing that there exist uncountably many distinct quasi-isometry classes of elementary amenable groups? How about solvable groups?

For amenable groups it follows from the result of Grigorchuk (proved in the 80's) stating that there are uncountably many groups of intermediate growth with pairwise incomparable growth functions.

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@Denis: to any f.g. group $G$, you can associate the set $N(G)$ of ultrafilters $\omega$ such that $Cone(G,\omega,(1/n))$ is a real tree. So $N(G)$ is a QI-invariant of $G$. I think the idea is to show that elementary amenable groups achieve continuum many sets $N(G)$. It's not obvious (and I haven't checked) but I guess you can do this. – YCor Dec 25 '11 at 22:33