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So the von Neumann problem talks about non-amenable groups without non-abelian free subgroups, but I'm just looking for the existence of a copy of $\mathbb{Z}$. Wouldn't this be easier?
There is active research going on about whether such problems are decidable if we assume the word problem has a solution. For instance, determining whether a group is a 3-manifold group is decidable, given a presentation and a solution to the word problem.
This is a very interesting proof! Unfortunately I am somewhat bound to use the Freiheitssatz in my exposition, but these stronger results are always interesting to note. Thank you!
