TITLE: Is there a finitely presented group with infinite homology over $\mathbb{Q}$?
QUESTION [8 upvotes]: 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?

REPLY [6 votes]: This answers the other part of your question, not answered by Thompson's group.  For each $i\geq 3$ there is a finitely presented group $G_i$ with the property that $H_i(G_i\mathbb{Q})$ is infinite dimensional.  The first such examples were due to Stallings ($i=3$ and Bieri $i>3$).  Take a direct product of $i$ copies of the free group $F$ on two generators, and define $G_i$ to be the kernel of the homomorphism $F^i\rightarrow \mathbb{Z}$ that sends each of the $2i$ standard generators to $1\in \mathbb{Z}$.  These are the Bieri-Stallings groups.  For $i=2$ the analogous group is finitely generated but not finitely presented, and for $i=1$ it is free of infinite rank.  These groups can be viewed as special cases of the Bestvina-Brady construction.