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Question 1 is interesting but, I'll guess, very difficult?

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 $K(G, 1)$. I suspect that the answers to 2 and 3 will remain "no"?

Question 1 is interesting but, I'll guess, very difficult?

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 $K(G, 1)$. I suspect that the answers to 2 and 3 will remain "no"?

As Igor points out (in the comments below) there are (torsion free) uniform lattices in the complex hyperbolic plane. The resulting quotients of the complex hyperbolic plane give compact four-manifolds without boundary which have word-hyperbolic fundamental groups. However, as Igor also points out, the complex hyperbolic plane is not quasi-isometric to real hyperbolic four-space. So the answer to 3 is "no" even with the aspherical assumption.

Question 1 is interesting but, I'll guess, very difficult?

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 $K(G, 1)$. I suspect that the answers to 2 and 3 will remain "no". For example, if I just knew a little bit about the complex hyperbolic plane...

Question 1 is interesting but, I'll guess, very difficult?

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 $K(G, 1)$. I suspect that the answers to 2 and 3 will remain "no"?

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 $K(G, 1)$.

Question 1 is interesting but, I'll guess, very difficult?

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 $K(G, 1)$. I suspect that the answers to 2 and 3 will remain "no". For example, if I just knew a little bit about the complex hyperbolic plane...