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.



Quasi-isometry classes of elementary amenable groups
The affirmative answer in given by Yves de Cornulier and Romain Tessera in arXiv:1203.4696.


