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!



amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?
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](http://www.ma.rhul.ac.uk/~uxah002/KrophollerMartinezNucinkis.pdf), but I am too busy right now to check. 


