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Dear Igor, could you be a bit more specific about the fact that Thurston was familiar with Deligne and Mostow's work when he first came up with the flat metric interpretation of Deligne-Mostow's orbifolds? Because obviously Thurston has been aware of their work at some point, but the interesting question is whether it influenced or inspired in some way his own work.
I get the fact that the geometrisation conjecture combined with the recent achievement on hyperbolic 3-manifolds gives a good picture of the exact complexity of the world 3-manifolds. But without this machinery, how good can be an approximation of this complexity ?
To me it is not obvious that I build this way a lot of different manifolds. I glue faces of a polyhedron and then get by Poincaré's theorem a presentation of the fundamental group of my manifold. But how can I quickly deduces that, for instance, this method builds infinitely many different manifolds ? Even if so, are these differentiated by their first Betti number ? Building this way only one homology sphere is (a little bit) painful and requires non-obvious computation on the fundamental group of the resulting manifold.
That is for sure another interesting construction. But the question I am asking is how can I convince myself that these constructions lead to non homeomorphic manifolds ? An how much of the combinatorial complexity of 3-manifolds can I catch this way ?
Dear Liviu, I'd be interested to know a bit more about your first comment. How do you catch the complexity of 3-manifolds from the complexity of links ? Why (just for the sake of the argument) this operation wouldn't lead to only a finite number of manifolds ? In other word why essentially different links give rise to two different manifolds ?
