# Is there a finitely presented group with infinite homology over $\mathbb{Q}$?

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?

• Just to make sure: the assertion that $H_i(G,\mathbb{Q})$ is finite-dimensional (for all $i>2$) is an assumption, yes? – Alex Suciu Aug 7 '14 at 4:24
• Also, "infinite homology" means "infinite-dimensional homology" (as $\mathbb{Q}$-vector space), right? – Alex Suciu Aug 7 '14 at 4:26
• @Alex: I thought any group homology of a finitely presented group is finite-dimensional. Are there examples where $H_2$ isn't? And yes, "infinite" means infinite-dimensional – Dmitry Vaintrob Aug 7 '14 at 8:07
• Yes, $H_2$ of an fp group is finitely generated. But, say, $H_3$ needs not be finitely generated. The first such example was given by John Stallings, in a seminal paper, titled, sure enough, A finitely presented group whose 3-dimensional integral homology is not finitely generated, see here. – Alex Suciu Aug 7 '14 at 8:27
• @Dmitry: you should edit the second sentence of your question according to Alex' comments. – YCor Aug 7 '14 at 9:54

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.