What are some examples of finitely generated (finitely presented) elementary amenable groups which are not virtually solvable?
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Houghton groups are (nonsplit) extensions of the group of the finitely supported permutations of the integers by $\mathbf{Z}^d$ for $d\ge 2$ and are thus elementary amenable and not virtually solvable. They're finitely presented, as shown by K.Brown here (Finiteness properties of groups, JPAA 1987). They were introduced by Houghton as f.g. groups with coset spaces with $2<n<\infty$ ends. 

