Groups $\pi$ with $K(\pi, 1)$ a finite CW-complex

Question

For what (say, finitely generated) groups $\pi$ is $K(\pi, 1)$ a finite CW-complex? Such a group must necessarily be torsion-free, since otherwise $H^*K(\pi, 1) = H^* \pi$ would be nontrivial in arbitrarily high dimensions. In the opposite direction, any non-positively curved manifold $X^n$ has $\exp:T_* X\to X$ a covering map by the Cartan-Hadamard theorem, and so $G = \pi_1 X$ works for reasonable such $X$. I've run across a couple of similar partial or ad-hoc results, but I think the general question is still unresolved. So, what's the current state of the question, and what are some neat partial results?

Answer 1

(For a reference for the following two definitions, see, for example, this paper.)

A group $G$ is said to be of type FL if $BG$ is homotopy equivalent to a finite CW complex, which is apparently equivalent to the condition that $\mathbb{Z}$ admits a finite free resolution as a $\mathbb{Z}[G]$-module. A group $G$ is said to be of type FP if $BG$ is a homotopy retract of a finite CW complex (the standard term for this is "finitely dominated"), which is apparently equivalent to the condition that $\mathbb{Z}$ admits a finite projective resolution as a $\mathbb{Z}[G]$-module.

In general, if a space is a homotopy retract of a finite CW complex, then there is a further obstruction to it being homotopy equivalent to a finite CW complex called Wall's finiteness obstruction. It's an open problem to determine whether this obstruction ever vanishes for classifying spaces $BG$, hence whether FP and FL are equivalent conditions on a group or not.

FP is a stronger version of the condition that $G$ is finitely presented (since it also implies that $G$ is torsion-free). It's probably hopeless to classify all such groups, in the same way that it's hopeless to classify all finitely presented groups, but you can at least ask for necessary and sufficient conditions, and probably if you search using these keywords you can find them in the literature.