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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 3-manifold 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 3-manifold from repeating surgeries on knots in $S^3$
@HJRW - Well, sort of. To show Dehn twists generate, it suffices to think about the "two-chain" 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 one-skeleton is co-compact. 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 3-manifold 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 3-manifold from repeating surgeries on knots in $S^3$
@Igor, ThiKu - I would say that the relation between Dehn twists and Dehn surgery requires a non-trivial 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 3-manifold 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 3-manifold 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 three-manifolds. 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 3-manifold 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 3-manifold from repeating surgeries on knots in $S^3$
added refs
Mar
21
revised Obtain any 3-manifold from repeating surgeries on knots in $S^3$
working
Mar
21
answered Obtain any 3-manifold 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