Editing Geometrization conjecture (section)

===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>.