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2d

revised 
Scotts Theorem for one ended Fuchsian groups
added missing word "regular". 
Mar 28 
revised 
Scotts Theorem for one ended Fuchsian groups
fixed typo 
Mar 28 
answered  Scotts Theorem for one ended Fuchsian groups 
Mar 28 
comment 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
However, that induction has a base case to worry about (Dehn deals directly with the case of the five holed sphere, as I recall) and the classification of surfaces is a simple sounding, yet deep, theorem.  Look, I am not saying that this theorem or its proof is super difficult. However, we should all remember that we have the great, great benefit of hindsight! That makes many things look easier than they actually are... 
Mar 28 
comment 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
@HJRW  Well, sort of. To show Dehn twists generate, it suffices to think about the "twochain" relation and to prove that the curve complex is connected (essentially). To show finite generation (and so finite generation by twists) we have to additionally check that vertex stabilizers are finitely generated and the action on the oneskeleton is cocompact. Now, you are going to say that the former is an easy induction. Also, you have already said that the latter is obvious (from the classification of surfaces). 
Mar 22 
comment 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
@Igor  The usual proof first proves that twists generate, and then proves that only finitely many twists suffice. 
Mar 22 
comment 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
@Igor, ThiKu  I would say that the relation between Dehn twists and Dehn surgery requires a nontrivial conceptual leap. For Igor's second question  yes the proof is easier if you don't care about finite generation. However Dehn does show that quadratically many twists (as measured in the genus) generate. Of course this is later improved by Lickorish and then by Humphries to linearly many twists. 
Mar 22 
comment 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
@Igor  Dehn did prove that Dehn twists generate the mapping class group  but that is just step two of the outline given above. 
Mar 22 
comment 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
@Igor  I know of no evidence that Dehn realized the connection between Dehn twists on a surface and integral Dehn surgeries along knots in threemanifolds. Stillwell asserts that Dehn did not have this idea  see the first paragraph of his introduction to his translation of Dehn's paper "Die Gruppe der Abbildungsklassen". 
Mar 22 
awarded  Nice Answer 
Mar 21 
comment 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
I thought about voting to close, but I then realized that this is really a reference request. As such it is appropriate for MO... 
Mar 21 
revised 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
added refs 
Mar 21 
revised 
Obtain any 3manifold from repeating surgeries on knots in $S^3$
working 
Mar 21 
answered  Obtain any 3manifold from repeating surgeries on knots in $S^3$ 
Mar 21 
comment 
hyperbolic metrics
You are welcome! If this is what you wanted, you should accept this answer. 
Mar 18 
revised 
hyperbolic metrics
added 65 characters in body 
Mar 18 
answered  hyperbolic metrics 
Mar 17 
revised 
Map of the Klein quartic from $CP^2$ to $R^3$
edited to make it shorter, and changed the symbol from \chi to \mathcal{X} 
Mar 15 
comment 
Are there CAT(1) spaces which are not trees whose Gromov boundary is disconnected?
This is homework; the question should be moved. As a hint  you should think of examples of CAT(1) spaces and their boundaries. Then think about how you can cut spaces into pieces (or glue spaces together) and how the boundary changes under those operations. 
Feb 13 
answered  History of Poincare conjecture in higher dimension 