A special residually finite group

Is there an example of a finitely generated (infinite) residually finite group $\Gamma$ for which every linear representation of $\Gamma$ has finite image?

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By linear representation, do you mean finite-dimensional linear representation? –  Yemon Choi Oct 10 '10 at 6:20
Yes, and over a field of characteristic zero. –  anon Oct 10 '10 at 6:31
I erased my answer, since it had the same properties as the Grigorchuk group (and is based on unpublished work), but I wanted to point out that Mark Sapir commented that the original construction of an infinite torsion residually finite group was due to Golod. en.wikipedia.org/wiki/Periodic_group –  Ian Agol Oct 11 '10 at 17:42
@IanAgol: did you mean some Wilson's construction of just infinte groups of type N(h)? Is it published now? –  Lev Glebsky Dec 14 '13 at 14:56

A group $G$ is just infinite if it is infinite but every proper quotient is finite.
@Andy: solvable groups can have exponential growth. For example ${\mathbb Z}\wr {\mathbb Z}$ is solvable of class 2 and has exponential growth because it contains a free non-cyclic subsemigroup. –  Mark Sapir Oct 10 '10 at 22:54