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 nonparallel curves on a surface, where each pair are allowed to intersect at most k times, is finite. 