Submitted to: Dr. Juliet Beringuel
Submitted by: Germaine L. Acapulco
Title of the Study: Groups of Piecewise Linear Homeomorphisms Author: Melanie Stein
Date Conducted: August 1991
In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups. Given a compact subinterval of the real line, the order preserving piecewise linear homeomorphisms of the interval form a group under composition. In this thesis we study subgroups of such groups obtained by restricting the slopes and singularities occurring in the homeomorphisms. We also consider analogous groups of homeomorphisms and bijections of the circle. This class of groups contains examples which have furnished the first examples of finitely presented infinite simple groups. We extend this by exhibiting simple subgroups in all of these groups, and showing them to be finitely presented in some cases. We then consider subgroups obtained by restricting slopes to a finitely generated subgroup of the rationals generated by integers, say n_1,n_2,...,n_k, and insisting that the singularities lie in Z[1 / (n_1 n_2...n_k)]. We construct contractible CW complexes on which these groups act with finite stabilizers, yielding classifying spaces as the quotients. We obtain combinatorial information about the groups by studying these classifying spaces. We compute homology and obtain finite presentations.