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On generalised Farey graphs and applications to the curve complex
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Collyer, Thomas P. A. (2012) On generalised Farey graphs and applications to the curve complex. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2683229~S1
Abstract
In the first part of the thesis, we introduce a family of simplicial complexes
called tree complexes, which generalise the well-known Farey graph. We study
numerous aspects of tree complexes. Firstly we show for a given dimension n, the
tree complex K(n) is simplicially rigid. We then study the geodesics between a
pair of given vertices x and y, giving a bound in terms of the distance between the
vertices, and showing that there always exist a pair of vertices at a given distance
which attains this bound. When n = 2, this bound is the ith Fibonacci number,
where i is the distance between the two vertices. We next study the automorphism
group of a tree complex, showing that it splits as a semi-direct product. Finally we
study the coarse geometry of a tree complex, showing in particular that for n > 2 each tree complex is quasi-isometric to the simplicial tree T [infinity].
In the second part of the thesis, we study the curve complex of the five-holed
sphere, C(S0,5), via subsurface projections to the four-holed sphere. We show that
geodesically embedded pentagons, hexagons and heptagons are unique, up to the
action of the mapping class group. We conjecture firstly that there are no larger
geodesically embedded cycles in C(S0,5), and secondly that these methods might be
used in a greatly simplified proof of the hyperbolicity of C(S0,5).
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Complexes, Graph theory, Series, Farey, Curves, Geodesics (Mathematics) | ||||
Official Date: | September 2012 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Bowditch, B. H. (Brian Hayward), 1961- | ||||
Sponsors: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Extent: | vii, 100 leaves : illustrations | ||||
Language: | eng |
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