Topological interpretation for groups of type $FP_2$ A group $G$ is of type $FP_2$ if it admits a partial projective resolution of $\mathbb{Z}G$-modules $$ P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb{Z} \rightarrow 0$$ with each $P_i$ being finitely generated and projective. 
In the event $G$ acts freely and cocompactly on a space $X$ with trivial first homology, then $G$ is of type $FP_2$. The converse holds if $G$ has cohomological dimension $2$, but does it hold in general?
 A: It is true that a group $G$ is $FP_2$ if and only if $G$ acts freely cellularly on a connected CW-complex with trivial first homology group.  I see from the comments that you are worried about finite generation of $G$: this is not a problem, because a group $G$ is $FP_1$ over any (non-trivial) ring if and only if $G$ is finitely generated.  In fact, the elements $g_1,...,g_n$ generate $G$ if and only if the elements $1-g_1,...,1-g_n$ generate the augmentation ideal of $\mathbb{Z}G$.  
Now suppose that $G$ is $FP_2$, and take an infinite presentation of $G$ with finitely many generators.  The Cayley graph for $G$ with respect to this generating set is a connected 1-dimensional complex with a free cocompact $G$-action.  Since $G$ is $FP_2$, the first homology group of this graph is finitely generated as a $\mathbb{Z}G$-module.  Now attach free orbits of 2-cells corresponding to the relators, one relator at a time.  If you attached all the relators you would get a 2-complex with trivial fundamental group.  Since the first homology of the graph you started with is finitely generated, you only need to attach finitely many of the orbits of cells to get a complex with trivial first homology.  This proves something slightly stronger: if you take the Cayley 2-complex for any presentation for $G$ with finitely many generators, then it has a $G$-invariant subcomplex that is cocompact and has trivial first homology.  
