  Let $G$ be a group with a finite balanced presentation $P$. We associate a
Heegaard Floer homology group $\widehat{HF}_P(G)$ with the pair $(G,P)$ based
on some extra choices and technical assumptions. We show that
$\widehat{HF}_P(G)$ is independent from these choices and also is invariant
under stable Andrews-Curtis transformations on $P$, based on two claims which
are not settled in this paper.
