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

$\begingroup$ By linear representation, do you mean finitedimensional linear representation? $\endgroup$ – Yemon Choi Oct 10 '10 at 6:20

$\begingroup$ Yes, and over a field of characteristic zero. $\endgroup$ – anon Oct 10 '10 at 6:31

$\begingroup$ I erased my answer, since it had the same properties as the Grigorchuk group (and is based on unpublished work), but I wanted to point out that Mark Sapir commented that the original construction of an infinite torsion residually finite group was due to Golod.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 $\endgroup$ – Ian Agol Oct 11 '10 at 17:42

$\begingroup$ @IanAgol: did you mean some Wilson's construction of just infinte groups of type N(h)? Is it published now? $\endgroup$ – Lev Glebsky Dec 14 '13 at 14:56
A group $G$ is just infinite if it is infinite but every proper quotient is finite.
Clearly a just infinite group which is not linear has the property that its image under any linear representation is finite. Thus any group which is finitely generated, residually finite, not linear and just infinite is an example of what you want: for instance, the Grigorchuk group.

3$\begingroup$ It's not hard to prove that the Grigorchuk grp is not linear. For instance, the Tits alternative says that every fg linear grp G either contains a solvable subgroup of finite index or contains a nonabelian free subgroup. This implies that the "growth function" f(n) of G (here f(n) is the number of elements of G of length at most n in a fixed genset) grows either polynomially (if G has a solvable subgrp of finite index) or exponentially (if G contains a nonabelian free subgrp). However, the first major thm about the Grigorchuk grp is that its growth fcn is superpolynomial but subexponential. $\endgroup$ – Andy Putman Oct 10 '10 at 20:54

1$\begingroup$ By the way, I highly recommend reading the final chapter in Pierre de la Harpe's book "Topics in Geometric Group Theory", which is entirely devoted to the Grigorchuk group. It serves as a sort of "universal counterexample" to conjectures in geometric group theory. $\endgroup$ – Andy Putman Oct 10 '10 at 21:09

1$\begingroup$ @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 noncyclic subsemigroup. $\endgroup$ – Mark Sapir Oct 10 '10 at 22:54

1$\begingroup$ @Pete, thanks for the far better rewriting of my answer. $\endgroup$ – Mustafa Gokhan Benli Oct 10 '10 at 23:59

1$\begingroup$ Andy  with a 'virtually' in front of 'nilpotent', that's the MilnorWolf Theorem. $\endgroup$ – HJRW Oct 11 '10 at 2:37