Editing Geometrization conjecture (section)

==The eight Thurston geometries==
A '''model geometry''' is a simply connected smooth manifold ''X'' together with a [[transitive action]] of a [[Lie group]] ''G'' on ''X'' with compact stabilizers.

A model geometry is called '''maximal''' if ''G'' is maximal among groups acting smoothly and transitively on ''X'' with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.

A '''geometric structure''' on a manifold  ''M'' is a [[diffeomorphism]] from ''M'' to  ''X''/Γ for some model geometry ''X'', where Γ is a [[Discrete group|discrete]] subgroup of ''G'' acting freely on ''X'' ; this is a special case of a complete [[(G,X)-structure|(''G'',''X'')-structure]]. If a given manifold admits a geometric structure, then it admits one whose model is maximal.

A 3-dimensional model geometry ''X'' is relevant to the geometrization conjecture if it is maximal  and if there is at least one compact manifold with a geometric structure modelled on ''X''. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called '''Thurston geometries'''. (There are also [[uncountably]] many model geometries without compact quotients.)

There is some connection with the [[Bianchi classification|Bianchi groups]]: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However  [[2-sphere|'''S'''<sup>2</sup>]] × '''R''' cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of [[solvable group|solvable]] [[Haar measure|non-unimodular]] Bianchi groups, most of which give model geometries with no compact representatives.

===Spherical geometry S<sup>3</sup>===
{{Main|Spherical geometry}}
The point stabilizer is O(3, '''R'''), and the group ''G'' is the 6-dimensional Lie group O(4, '''R'''), with 2 components. The corresponding manifolds are exactly the closed 3-manifolds with finite [[fundamental group]]. Examples include the [[3-sphere]], the [[Poincaré homology sphere]], [[Lens space]]s.  This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type IX]]. Manifolds with this geometry are all compact, orientable, and have the structure of a [[Seifert fiber space]] (often in several ways). The complete list of such manifolds is given in the article on [[spherical 3-manifold]]s. Under Ricci flow, manifolds with this geometry collapse to a point in finite time.

===Euclidean geometry ''E''<sup>3</sup>===
{{Main|Euclidean geometry}}
The point stabilizer is O(3, '''R'''), and the group ''G'' is the 6-dimensional Lie group '''R'''<sup>3</sup> × O(3, '''R'''), with 2 components. Examples are the [[three-torus|3-torus]], and more generally the [[mapping torus]]  of a finite-order [[automorphism]] of the 2-torus; see [[torus bundle]]. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi groups of type I or VII<sub>0</sub>]]. Finite volume manifolds with this geometry are all compact,  and have the structure of a [[Seifert fiber space]] (sometimes in two ways). The complete list of such manifolds is given in the article on [[Seifert fiber space]]s. Under Ricci flow, manifolds with Euclidean  geometry  remain invariant.

===Hyperbolic geometry H<sup>3</sup>===
{{Main|Hyperbolic geometry}}
The point stabilizer is O(3, '''R'''), and the group ''G'' is the 6-dimensional Lie group O<sup>+</sup>(1, 3, '''R'''), with 2 components. There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest volume is the [[Weeks manifold]]. Other examples are given by the [[Seifert–Weber space]], or "sufficiently complicated" [[Dehn surgery|Dehn surgeries]] on [[link (knot theory)|link]]s, or most [[Haken manifold]]s. The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, [[atoroidal]], and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type V or VII<sub>h≠0</sub>]]. Under Ricci flow, manifolds with hyperbolic  geometry expand.

===The geometry of S<sup>2</sup> × R===
The point stabilizer is O(2, '''R''') × '''Z'''/2'''Z''', and the group ''G'' is O(3, '''R''') × '''R''' × '''Z'''/2'''Z''', with 4 components.  The four finite volume manifolds with this geometry are: '''S'''<sup>2</sup> × '''S'''<sup>1</sup>, the mapping torus of  the antipode map of '''S'''<sup>2</sup>, the connected sum of two copies of 3-dimensional projective space, and the product of '''S'''<sup>1</sup> with two-dimensional projective space. The first two are mapping tori of  the identity map and antipode map of the 2-sphere, and are the only examples of 3-manifolds that are prime but not irreducible. The third is the only example of a non-trivial connected sum with a geometric structure. This is the only model geometry that cannot be realized as a left invariant metric on a 3-dimensional Lie group. Finite volume manifolds with this geometry are all compact and have the structure of a [[Seifert fiber space]] (often in several ways). Under normalized Ricci flow manifolds with this  geometry converge to a 1-dimensional manifold.

===The geometry of H<sup>2</sup> × R===
The point stabilizer is O(2, '''R''') × '''Z'''/2'''Z''', and the group ''G'' is O<sup>+</sup>(1, 2, '''R''') × '''R''' × '''Z'''/2'''Z''', with 4 components. Examples include the product of a [[hyperbolic surface]] with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface. Finite volume manifolds with this geometry  have the structure of a [[Seifert fiber space]] if they are orientable. (If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may "reverse orientation";  in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori.<ref>{{cite journal |first=Ronald |last=Fintushel |title=Local S<sup>1</sup> actions on 3-manifolds |journal=[[Pacific Journal of Mathematics]] |volume=66 |number=1 |year=1976 |pages=111–118  | doi=10.2140/pjm.1976.66.111 |doi-access=free }}</ref>) The classification of such (oriented) manifolds is given in the article on [[Seifert fiber space]]s. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type III]]. Under normalized Ricci flow manifolds with this  geometry converge to a 2-dimensional manifold.

===The geometry of the universal cover of SL(2, R)===
The [[universal cover]] of [[SL2(R)|SL(2, '''R''')]] is denoted <math>{\widetilde{\rm{SL}}}(2, \mathbf{R})</math>.  It fibers over '''H'''<sup>2</sup>, and the space is sometimes called "Twisted H<sup>2</sup> × R". The group ''G'' has 2 components. Its identity component has the structure <math>(\mathbf{R}\times\widetilde{\rm{SL}}_2 (\mathbf{R}))/\mathbf{Z}</math>.  The point stabilizer is O(2,'''R''').

Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the [[homology sphere|Brieskorn homology spheres]] (excepting the 3-sphere and the [[Poincaré dodecahedral space]]). This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type VIII or III]]. Finite volume manifolds with this geometry  are orientable and have the structure of a [[Seifert fiber space]]. The classification of such manifolds is given in the article on [[Seifert fiber space]]s. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.

===Nil geometry===
{{see also|Nilmanifold}}

This fibers over ''E''<sup>2</sup>, and so is sometimes known as "Twisted ''E''<sup>2</sup> × R". It is the geometry of the [[Heisenberg group]]. The point stabilizer is O(2, '''R'''). The group ''G'' has 2 components, and is a semidirect product of the 3-dimensional Heisenberg group by the group O(2, '''R''') of isometries of a circle. Compact manifolds with this geometry include the mapping torus of  a [[Dehn twist]] of a 2-torus, or the quotient of the Heisenberg group by the "integral Heisenberg group". This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type II]]. Finite volume manifolds with this geometry  are compact and orientable and have the structure of a [[Seifert fiber space]]. The classification of such manifolds is given in the article on [[Seifert fiber space]]s. Under normalized Ricci flow, compact manifolds with this  geometry converge to '''R'''<sup>2</sup> with the flat metric.

===Sol geometry===
{{see also|Solvmanifold}}

This geometry (also called '''Solv geometry''') fibers over the line with fiber the plane, and is the geometry of the identity component of the group ''G''. The point stabilizer is the dihedral group of order 8. The group  ''G'' has  8 components, and is the group of maps from 2-dimensional Minkowski space to itself that are either isometries or multiply the metric by −1. The identity component  has a normal subgroup '''R'''<sup>2</sup> with quotient '''R''', where '''R''' acts on '''R'''<sup>2</sup> with 2 (real) eigenspaces, with distinct real  eigenvalues of product 1. This is the  [[Bianchi classification|Bianchi group of type VI<sub>0</sub>]] and the geometry can be modeled as a left invariant metric on this group. All finite volume manifolds with solv geometry are compact. The compact  manifolds with solv geometry are either the [[mapping torus]] of  an [[Anosov map]] of the 2-torus (such a map is an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as <math>\left( {\begin{array}{*{20}c}
   2 & 1  \\
   1 & 1  \\
\end{array}} \right)</math>), or quotients of these by groups of order at most 8. The eigenvalues of the automorphism of the torus generate an order of a real quadratic field, and the solv manifolds can be classified in terms of the units and ideal classes of this order.<ref>{{Cite journal |last1=Quinn |first1=Joseph |last2=Verjovsky |first2=Alberto |date=2020-06-01 |title=Cusp shapes of Hilbert–Blumenthal surfaces |url=https://doi.org/10.1007/s10711-019-00474-w |journal=Geometriae Dedicata |language=en |volume=206 |issue=1 |pages=27–42 |doi=10.1007/s10711-019-00474-w |s2cid=55731832 |issn=1572-9168|arxiv=1711.02418 }}</ref>
Under normalized Ricci flow compact manifolds with this  geometry converge (rather slowly) to '''R'''<sup>1</sup>.