Hyperbolic geometry, regular representations and curves on surfaces
Doctor of Philosophy
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.