Geometrization Conjecture - Uniqueness

Uniqueness

A closed 3-manifold has a geometric structure of at most one of the 8 types above, but finite volume non-compact 3-manifolds can occasionally have more than one type of geometric structure. (However a manifold can have many different geometric structures of the same type; for example, a surface of genus at least 2 has a continuum of different hyperbolic metrics.) More precisely, if M is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π₁(M):

• If π₁(M) is finite then the geometric structure on M is spherical, and M is compact.
• If π₁(M) is virtually cyclic but not finite then the geometric structure on M is S2×R, and M is compact.
• If π₁(M) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.
• If π₁(M) is virtually nilpotent but not virtually abelian then the geometric structure on M is nil geometry, and M is compact.
• If π₁(M) is virtually solvable but not virtually nilpotent then the geometric structure on M is sol geometry, and M is compact.
• If π₁(M) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H2×R or the universal cover of SL(2, R). The manifold M may be either compact or non-compact. If it is compact, then the 2 geometries can be distinguished by whether or not π₁(M) has a finite index subgroup that splits as a semidirect product of the normal cyclic subgroup and something else. If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples (such as the complement of a trefoil knot) where a manifold may have a finite volume geometric structure of either type.
• If π₁(M) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.

Infinite volume manifolds can have many different types of geometric structure: for example, R3 can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it. Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.

There can be more than one way to decompose a closed 3-manifold into pieces with geometric structures. For example:

• Taking connected sums with several copies of S3 does not change a manifold.
• The connected sum of two projective 3-spaces has a S2×R geometry, and is also the connected sum of two pieces with S3 geometry.
• The product of a surface negative curvature and a circle has a geometric structure, but can also be cut along tori to produce smaller pieces that also have geometric structures. There are many similar examples for Seifert fiber spaces.

It is possible to choose a "canonical" decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori. However this minimal decomposition is not necessarily the one produced by Ricci flow; if fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.