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Sam Nead
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Question 1 is interesting but, I strongly suspect, very hard. I'd be interested in other folks' thoughts.

The answer to question 2 is "no". This is because any finitely presented group appears as the fundamental group of some compact four-manifold (without boundary). I suppose that I also need to display an infinite collection of quasi-isometry types of finitely presented groups so... $\mathbb{Z}^n$.

The answer to question 3 is "no". Consider $S^2 \times S_2$, the two-sphere crossed with the surface of genus two.

Perhaps you want to add the hypothesis that the four-manifold is aspherical: that is, the four-manifold is a $K(G, 1)$.

Sam Nead
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