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Yes, there are conditions; basically you want to maintain the hypothesis of not having incompressible annuli at each stage. A really good reference for this is Morgan's essay, "On Thurston's uniformiz …

I'm assuming that your intention is that negatively curved means having negative sectional curvature. Your question, with regard to uniqueness up to homeomorphism, is a special case of the Borel Conj …

Here's another negative answer for Q2. I'm assuming (as in Sam Nead's answer) that the covering should be locally isometric. By Ahlfors-Bers, a tame infinite volume hyperbolic manifold with ends of th …

I'm happy to be able to answer my own question! John Ratcliffe, Steven Tschantz and I showed that the Dirac operator on the Davis manifold (a closed hyperbolic 4-manifold constructed by Mike Davis) ha …

If you want to know if your group is not discrete, apply Jorgensen's inequality (Jørgensen, Troels (1976), "On discrete groups of Möbius transformations", American Journal of Mathematics 98 (3): 739–7 …

Ted Shifrin's course notes on curves and surfaces has a nice introduction to hyperbolic geometry in the plane. You have to refer to the earlier part of the book for the notation related to Riemannian …

For question 2, the answer is no. There are plenty of distinct hyperbolic knots (and hence having distinct fundamental group) with the same volume. For example, the Conway and Kinoshita-Terasaka knot …

With regard to surfaces: it is a simple theorem that $\pi_2$ of any closed $4$-manifold is torsion free; so one doesn't need any complex geometry to see that there are no such projective surfaces. Th …