Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H\_i(G, \mathbb{Q})$, or equivalently $H\_i(BG, \mathbb{Q})$ (topological homology of the classifying space) be infinite-dimensional? Can they be nonzero for infinitely many $i$?


For any finitely presented groups I've seen, the answer is a surprising "no" (all finitely presented groups I know act on a finite-dimensional contractible space with finite stabilizers, and it follows that above the dimension of this space, homology vanishes). But it really should be the case that a "general" finitely-presented group has infinite homology... does anyone know of an example?



Is there a finitely presented group with infinite homology over $\mathbb{Q}$?
Thompson's group F is an example. It's finitely presented and, according to [this](http://www.math.cornell.edu/~kbrown/papers/homology.pdf) paper of Ken Brown, the integral homology is free abelian of rank 2 in every positive dimension. 


