What are some examples of finitely generated (finitely presented) elementary amenable groups which are not virtually solvable?
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9$\begingroup$ Take the group of permutations with finite support of $\mathbb{Z}$: this group is non-virtually solvable, and locally finite (hence elementary amenable); of course it is not finitely generated. To make it f.g., take the semi-direct product by $\mathbb{Z}$ acting by conjugation by the shift. $\endgroup$– Alain ValetteCommented Sep 24, 2012 at 18:52
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3$\begingroup$ Houghton groups are analogues of these, but are in addition finitely presented. $\endgroup$– YCorCommented Sep 25, 2012 at 8:40
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3$\begingroup$ Apparently, the question on whether there exists a finitely presented elementary amenable group which is not virtually solvable appears in Kourovka notebook (asked by Linnell-Schick). Linnell mentioned it to me last summer and I also immediately thought of Houghton groups (as I had just read the paper by Brown math.cornell.edu/~kbrown/scan/1987.0044.0045.pdf where their finite presentability was established). $\endgroup$– Igor BelegradekCommented Sep 25, 2012 at 11:54
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Houghton groups are (non-split) 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.