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can you give me an example of a 2 generated nonlinear solvable group? All metabelian groups are linear due to Magnus. An example of a 3 generated nonlinear solvable group is $(\mathbb{Z} \wr \mathbb{Z})\wr \mathbb{Z}$.

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I don't think I can answer your question, but out of curiosity, what does $\wr$ denote? –  tomasz Aug 22 '12 at 22:25
    
@tomasz , that is the usual notation for the (regular) wreath product –  DonAntonio Aug 23 '12 at 0:03

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up vote 5 down vote accepted

There are even non-residually finite 2-generated solvable groups (of class 3). See Denis Osin's answer here: http://mathoverflow.net/questions/30653/hnn-embedding-theorem-for-amenable-groups .

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