TITLE: amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?
QUESTION [9 upvotes]: I heard from someone that the following problem is an open question.
(Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,y|xy^mx^{-1}=y^n\rangle$ and it admits a finite $K(G,1)$, is it a hyperbolic group?
I could not find the relevent stuff on this problem, so I am wondering whether the following is known.
(My question)For a countable discrete amenable group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,y|xy^mx^{-1}=y^n\rangle$ and it admits a finite $K(G,1)$, is it a virtually cyclic group?
Note that ``Yes for problem 1 implies Yes for my question" since $G$ is amenable and hyperbolic iff $G$ is virtually cyclic(?)
Any references or comments are welcome!

REPLY [7 votes]: Yes, this problem is as open as Problem 1. There are no feasible obstructions to counter-examples, the main obstacle is lack of tools: All known constructions of non-hyperbolic groups with finite $K(G,1)$, yield groups which contain some $BS(p,1)$, $p\ge 1$. 
Edit: For elementary amenable groups of finite type, positive answer to your question probably follows from the results here, but I am too busy right now to check.