dc.contributor.advisor Gadgil, Siddhartha dc.contributor.author Sanki, Bidyut dc.date.accessioned 2018-01-31T05:17:04Z dc.date.accessioned 2018-07-31T06:09:12Z dc.date.available 2018-01-31T05:17:04Z dc.date.available 2018-07-31T06:09:12Z dc.date.issued 2018-01-31 dc.date.submitted 2014 dc.identifier.uri https://etd.iisc.ac.in/handle/2005/3049 dc.identifier.abstract http://etd.iisc.ac.in/static/etd/abstracts/3913/G26900-Abs.pdf en_US dc.description.abstract 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). en_US dc.language.iso en_US en_US dc.relation.ispartofseries G26900 en_US dc.subject Hyperbolic Surfaces en_US dc.subject Systolic Graphs en_US dc.subject Fat Graphs en_US dc.subject Geodesics en_US dc.subject Hyperbolic Geometry en_US dc.subject Graphic Methods en_US dc.subject Polygonal Quasi-Geodesics en_US dc.subject Quasi-geodesics en_US dc.subject Gauss-Bonnet Theorem en_US dc.subject Hyperbolic Trigonometry en_US dc.subject Graphs en_US dc.subject Trivalent Graphs en_US dc.subject.classification Mathematics en_US dc.title Shortest Length Geodesics on Closed Hyperbolic Surfaces en_US dc.type Thesis en_US dc.degree.name PhD en_US dc.degree.level Doctoral en_US dc.degree.discipline Faculty of Science en_US
﻿