Some classmates and I were working on the following question - is the fundamental group of the Klein Bottle $K$ torsion-free? We have the following presentation: $$\pi_1(K) = \langle a,b: aba = b \rangle.$$
In trying to answer this problem, we came up with the following question that could resolve this problem, and seemed like an interesting claim in general:
Is the Fundamental Group of a space with contractible universal cover torsion-free?
We thought about it a bit and unfortunately did not find a counterexample or a proof. Is this true? If it's not, what is a good way to answer our original question about the Klein Bottle?