Author's School

Graduate School of Arts & Sciences

Author's Department/Program

Mathematics

Language

English (en)

Date of Award

January 2009

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Chair and Committee

Rachel Roberts

Abstract

Thurston and, later, Calegari-Dunfield found superlaminations in certain laminated 3-manifolds, the existence of which implies inclusions into Homeo S1 of the fundamental groups of those manifolds. The present paper extends the construction of the superlamination, and finds an infinite class of manifolds to which the extension does not yield such an inclusion of groups. Specifically, Calegari and Dunfield's proof of the existence of such an inclusion into Homeo S1 depended on their filling lemma, which states that essential laminations with solid torus guts can have leaves added to them to yield essential laminations with solid torus complementary regions.: Roughly, a gut is that part of a complementary region that is not sandwiched between only two leaves.) The present paper finds the leafspace of the resultant lamination, and extends Calegari and Dunfield's operation to more general cases: first to reduce any finite-genus-handlebody complementary region to its gut, and then to reduce the genus of a complementary region even where doing so modifies the gut itself. In these cases, too, then, there can be an inclusion of the manifold's fundamental group into Homeo S1. Cataclysms correspond to non-Hausdorffness in the leafspace of a lamination. A cataclysm is orderable if some order on it is invariant under deck transformations. Calegari-Dunfield showed that orderability of cataclysms, which is weaker than Hausdorffness of the leafspace, is sufficient for the existence of an inclusion into Homeo S1. The present paper finds a criterion for the non-orderability of cataclysms, and a class of examples satisfying the criterion.

Comments

Permanent URL: http://dx.doi.org/10.7936/K79021VD

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