Talk:Geometrization conjecture

message: for the non-specialist it would enhance much the article as a similar work that came in sciam 'poncaire conjeture' a few years ago  — Preceding unsigned comment added by 31.4.58.20 (talk) 19:46, 29 May 2015 (UTC)
thread_title:  drawings please 

message: Did they really offer him the prizes for a sketch of a proof? Is this a fair wording?  — Preceding unsigned comment added by 143.215.125.141 (talk) 22:16, 19 November 2013 (UTC)
thread_title:  Perelman "sketched" the proof? 

message: This may be partly because the standard two-level decomposition article isn't written yet, but I find the description of this phenomenon given here to be ambiguous.
Does it mean that each compact 3-manifold can be decomposed into prime 3-manifolds, each of which can be further decomposed via the Jaco-Shalen-Johannson torus decomposition?
:Yes, that's right.  See irreducible for a description of the prime decomposition.  The JSJ decomposition is applied to each prime manifold except for certain exceptions (since the JSJ decomposition requires irreducibility and every prime manifold, except for these exceptions are irreducible).  
:Yeah, someone should write something on the JSJ decomposition, and the prime decomposition also, although I've somewhat abated the problem by writing some stuff under irreducible.
Does it mean that each compact 3-manifold can be decomposed into prime 3-manifolds OR via the the Jaco-Shalen-Johannson torus decomposition?
Does it mean that each compact 3-manifold can be decompsed in two quite different ways, into prime 3-manifolds AND via the Jaco-Shalen-Johannson torus decomposition?
:It means, first decompose along certain essential spheres to get prime pieces and cap off sphere boundaries with three-balls.  THEN decompose along certain essential tori to get the JSJ decomposition.  By the way -- I've never heard anyone use the term "two-level" decomposition, so I am getting rid of it. Sam nead 23:50, 5 September 2006 (UTC)
thread_title: Two-level??

message: The description of sol geometry given here differs from that on MathWorld.
:Yeah, it doesn't make any sense.  Anyway, I removed the error, but at some point I will try and write an article.  I've added some references for now, in case anybody wants to look at them or write some stuff.  --C S (Talk) 08:33, 11 June 2006 (UTC)
thread_title:  Sol Geometry 

message: I wanted to make a few clarifying remarks about why the article mentions both these terms.  They are both common, although I've never considered which would be more common.  My initial revert was due to the fact that in my experience, when the hyp. theorem is explained, usually one starts "Let M be an atoroidal, blah blah" and ends with "M is hyperbolic", whereas the geometrization theorem's statement utilizes all 8 geometries.  Of course, it's simple to go from one to the other, which is presumably why the names appear to be basically synonymous (at least from my cursory check).
Issues of common usage aside (this can be explained however others feel is ok in the article, names can be reordered if needed, etc.), both names should be mentioned as geometrization theorem redirects here.  A common source of confusion after Perelman's work on GC has been some non-specialists and laymen calling GC the geometrization theorem, which as far as I know, is not at all that common to do so.  "Geometrization theorem" has a long history behind it, starting with Thurston himself, so the name, unfortunately, is taken.  I suppose people were never really concerned with this as it seemed highly unlikely GC would be proven in our lifetimes.  One day  "geometrization theorem" may have its own article (possibly under the name "hyperbolization theorem"), but at the moment, it doesn't.  So the redirect and mention in this article is the workaround.
Also, I changed "hyperbolisation" to "hyperbolization" for style consistencies.  I hope I haven't offended those in the Commonwealth. ^_^  --C S (Talk) 21:28, 21 April 2007 (UTC)
replies: text: I agree with your basic motivation but the specific "geometrization theorem" might be confusing. I'm curious where you see geometrization theorem occur in the literature.  All the books on the topic tend to call it the hyperbolization theorem: Otal, Kapovich, Hubbard.. etc.  On top of that, wasn't it Peter Scott that spelt out all the details regarding geometrization of Seifert fibred-spaces? This is basically the only difference between 'hyperbolisation' and 'geometrization' in the Haken case anyhow, except for Sol manifolds.  I think it's clear Thurston knew these details extremely well, but did he write them up? I haven't looked at his BAMS (1982) article in a while so the details are foggy. The only other article of Thurston's that touches the topic is his notes that got turned into his book, but it doesn't actually go into any Seifert-fibred details. Rybu 09:42, 22 April 2007 (UTC)
replies: text: Where?  Well, you can try Google Scholar and MathSciNet; you'll immediately get a number of relevant hits.  --C S (Talk) 10:08, 22 April 2007 (UTC)
thread_title:  geometrization theorem vs. hyperbolization theorem 

message: A basic search in the Front for the arXiv produce the numbers of articles:
*Solv geometry: 622, and solv manifold: 210
vs
*Sol geometry: 55, and sol manifold: 31
going on with the majority it should be used Solv. Thruston-self uses the term
solvegeometry.--kiddo 22:26, 16 July 2007 (UTC)
thread_title: Sol or Solv?

message: what is a "point stabilizer"?  —Preceding unsigned comment added by 74.170.68.234 (talk) 13:05, 17 March 2008 (UTC)
thread_title: ???

message: Many of the citations seem to be private lecture notes, Arxiv preprints and other non-reliable sources.  Could we have some peer-reviewed versions please?  Julian Birdbath (talk) 18:54, 15 March 2011 (UTC)
replies: text: Hatcher's notes are a well-known and widely-acknowledged freely available resource.  It counts as reliable.  The arxiv preprints are part of Grigori Perelman's solution to the Poincare conjecture, and are considered to be some of the most important mathematical works in the last 100 years.  Complaining about them is ludicrous in the extreme.  Sławomir Biały (talk) 14:00, 16 March 2011 (UTC)
thread_title:  Sources 

message: I find that this article is nearly indecipherable to someone who has not taken some university course that addresses the subject. If there is an expert in the field willing to make this article more understandable to non-experts, this would be ideal to wikipedia's encyclopedic mode.  — Preceding unsigned comment added by Azmisov (talk • contribs) 17:11, 27 December 2012 (UTC)
thread_title:  Technicality 

message: The article defines a geometry as
:[...] a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers.
but then speaks about geometric structures of "finite volume". So: What is "finite volume", starting from that definition? Or perhaps: Is a geometric structure a special case of a metric? If so, how does the metric on X arise? From some left-invariant metric on the Lie group G? If not: Is "volume" defined? Or only "finite/infinite volume"? --46.127.232.215 (talk) 09:47, 8 July 2014 (UTC)
thread_title:  Does it come with a metric? 

message: :
That's a bit stronger than the last version I saw (possibly in Thurston's book), which allows cuts by spheres and Klein bottles as well as tori. —Tamfang (talk) 05:01, 25 November 2015 (UTC)
thread_title:  kinds of cuts