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Covering maps and hulls in the curve complex
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Tang, Robert (2013) Covering maps and hulls in the curve complex. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2690211~S1
Abstract
This thesis studies the coarse geometry of the curve complex using intersection
number techniques. We show how weighted intersection numbers can be studied
using appropriate singular Euclidean surfaces. We then introduce a coarse analogue
of the convex hull of a finite set of vertices in the curve complex, called the short curve
hull, and provide intersection number conditions to find nearest point projections
to such hulls. We also obtain an upper bound for distances in the curve complex
using a greedy algorithm due to Hempel.
Covering maps between surfaces also play a significant part in this thesis.
We give a new proof of a theorem of Rafi and Schleimer which states that a covering
map between surfaces induces a natural quasi-isometric embedding between
their corresponding curve complexes. Our proof employs a distance estimate via a
suitable hyperbolic 3-manifold which arises from work on the proof of the Ending
Lamination Theorem. We then define an operation using a given covering map and
intersection number conditions and show that it approximates a nearest point projection
to the image of Rafi-Schleimer's map. We also prove that this operation
approximates a circumcentre of the orbit of a vertex in the curve complex under the
deck transformation group of a regular cover
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Curves, Algebraic, Number theory, Transformation groups, Intersection theory | ||||
Official Date: | June 2013 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Bowditch, Brian | ||||
Sponsors: | University of Warwick. Mathematics Institute; Warwick Postgraduate Research Scholarship | ||||
Extent: | vi, 65 leaves | ||||
Language: | eng |
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