Editing Geometrization conjecture (section)

===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.