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Shortest Length Geodesics on Closed Hyperbolic Surfaces

Electronic Theses of Indian Institute of Science

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Title Shortest Length Geodesics on Closed Hyperbolic Surfaces
 
Creator Sanki, Bidyut
 
Subject Hyperbolic Surfaces
Systolic Graphs
Fat Graphs
Geodesics
Hyperbolic Geometry
Graphic Methods
Polygonal Quasi-Geodesics
Quasi-geodesics
Gauss-Bonnet Theorem
Hyperbolic Trigonometry
Graphs
Trivalent Graphs
Mathematics
 
Description Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface -we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics.
Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
 
Contributor Gadgil, Siddhartha
 
Date 2018-01-31T05:17:04Z
2018-01-31T05:17:04Z
2018-01-31
2014
 
Type Thesis
 
Identifier http://hdl.handle.net/2005/3049
http://etd.ncsi.iisc.ernet.in/abstracts/3913/G26900-Abs.pdf
 
Language en_US
 
Relation G26900