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Maloni, Sara (2013) Slices of quasifuchsian space. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b2689287~S1
Abstract
In Chapter 1 we present the background material about curves on surfaces. In particular
we define the DehnThurston coordinates for the set S = S(Σ) of free homotopy class
of multicurves on the surface Σ. We also prove new results, like the precise relationship
between Penner's and D. Thurston's definition of the twist coordinate and the formula for
calculating the Thurston's symplectic form using DehnThurston coordinates.
For Chapter 2, let Σ be a surface of negative Euler characteristic together with a pants
decomposition PC. Kra's plumbing construction endows Σ with a projective structure as
follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb',
adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across
the ith pants curve is denied by a complex parameter μi ∈ C. The associated holonomy
representation ρμ : π1 (Σ)> PSL(2;C) gives a projective structure on Σ which depends
holomorphically on the μi. In particular, the traces of all elements ρ
μ (γ), where
γ ∈ π1 (Σ),
are polynomials in the μi.
Generalising results proved in [24; 40] for the once and twice punctured torus respectively,
we prove in Chapter 2 a formula giving a simple linear relationship between the coefficients
of the top terms of Tr ρμ (λ
), as polynomials in the μi, and the DehnThurston coordinates
of
relative to PC. We call this formula the Top Terms' Relationship.
In Chapter 3, applying the Top Terms' Relationship, we determine the asymptotic directions
of pleating rays in the Maskit embedding of a hyperbolic surface Σ as the bending
measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit
embedding M of a surface Σ is the space of geometrically finite groups on the boundary
of Quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom'
end consists of triply punctured spheres, the remains of Σ when the pants curves have been
pinched. Given a projective measured lamination [η] on Σ, the pleating ray P = P[η] is the
set of groups in M for which the bending measure pl+(G) of the top component ∂C+ of the
boundary of the convex core of the associated 3manifold H3=G is in the class [η].
Item Type:  Thesis (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Curves on surfaces, Euler characteristic, Algebraic spaces  
Official Date:  January 2013  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Series, Caroline  
Extent:  xvii, 124 leaves  
Language:  eng 
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