On Groups of PL-homeomorphisms of the Real Line

TL;DR

This paper studies generalized groups of piecewise linear homeomorphisms of the real line, extending Thompson's group F, and investigates their properties, isomorphisms, and automorphism groups.

Contribution

It introduces a parametric family of groups generalizing Thompson's F and provides complete classifications of their isomorphisms and automorphisms under certain conditions.

Findings

Determined when these groups are isomorphic.

Analyzed the automorphism groups of the generalized groups.

Established properties of these groups in relation to Thompson's original group.

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.