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Bell, Mark Christopher (2015) Recognising mapping classes. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2861353~S1
Abstract
This thesis focuses on three decision problems in the mapping class groups of surfaces, namely the reducibility, pseudo-Anosov and conjugacy problems. For a fixed surface, we use ideal triangulations to model both its mapping class group and space of measured laminations. This allows us to state these problems combinatorially. We give new solutions to each of these problems that, unlike the existing solutions which are based on the Bestvina–Handel algorithm, run in polynomial time when given a suitable certificate. This allows us to show that in fact each of these problems lies in the complexity class NP∩ co-NP instead of just EXPTIME.
At the heart of each of our solutions is the maximal splitting sequence of a projectively invariant measured lamination, as described by Agol. The complexity of this sequence bounds the difficulty of determining many of the properties of such a lamination, including whether it is filling. In Chapter 4 we give explicit polynomial upper bounds on the periodic and preperiodic lengths of such a sequence. This allows us to construct the running time bounds needed to show that these problems lie in NP∩co-NP.
We finish with a discussion of an implementation of these algorithms as part of the Python package flipper. We include several examples of properties of mapping classes that can be computed using it.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Class groups (Mathematics), Mappings (Mathematics), Group theory | ||||
Official Date: | June 2015 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Schleimer, Saul | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Extent: | xv, 133 leaves : illustrations (chiefly colour) | ||||
Language: | eng |
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