This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is instead automatic and contains no subgroups isomorphic to $\mathbb{Z} \times \mathbb{Z}$? If the first question is open, can someone clarify what is meant by "admit finite $k(G, 1)$" (is $G$ the fundamental group of a space with trivial higher homotopy groups or is the action of $G$ on the space properly discontinuous)? Any helpful references would be great.
