On Groups of PL-homeomorphisms of the Real Line

Abstract

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G are studied which depend on three parameters I, A, and P and which generalize the local definition of Thompson's group F thus: G consists of all orientation preserving PL-homeomorphisms of the real line with supports in the interval I, slopes in the multiplicative subgroup P of the positive reals and breaks in a finite subset of the additive P submodule A of R. A first aim of the monograph is to investigate in which form familiar properties of F continue to hold for these groups. Main aims of the monograph are the determination of isomorphisms among the groups G and the study of their automorphism groups. Complete answers are obtained if the group P is not cyclic or if the interval I is the full line.