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Hyperbolic geometry, regular representations and curves on surfaces

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Title: Hyperbolic geometry, regular representations and curves on surfaces
Author: Dona, Luca
Abstract: Tools and techniques in hyperbolic geometry are developed and applied primarily to questions about intersections of curves on surfaces. Formulae which explicitly relate the coefficients of a matrix to the geometric data of a hyperbolic transformation are found and applied. A purely algebraic general criterion for an element to be simple is found. Particularly convenient representations of Fuchsian groups are discussed. These representations have coefficients which belong to a ring with integral coefficients, and have nice symmetry properties. As a result, an algorithm to recover the word of a matrix in terms of the generating elements is given. Also, the general criterion for simplicity previously found is further reduced to a system of diophantine equations. In the appendices, another purely combinatorial algorithm for loops on surfaces is given, as well as a short proof of Dehn's solution to the word problem, and a proof that a system of simple non-parallel curves on a surface, where each pair are allowed to intersect at most k times, is finite.
Citation: Dona, Luca. (1995) "Hyperbolic geometry, regular representations and curves on surfaces." Doctoral Thesis, Rice University. http://hdl.handle.net/1911/16814.
URI: http://hdl.handle.net/1911/16814
Date: 1995

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