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Title: Recognising mapping classes
Author: Bell, Mark Christopher
ISNI:       0000 0004 5915 1479
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2015
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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.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics