Groups of PL homeomorphisms of cubes

TL;DR

This paper investigates algebraic properties of groups of PL and smooth homeomorphisms of cubes and manifolds, revealing their local indicability or circular orderability and analyzing their algebraic constraints and examples.

Contribution

It establishes new algebraic properties of these groups, such as local indicability and circular orderability, and provides examples and constraints for their actions.

Findings

Groups of PL homeomorphisms of the n-cube are locally indicable.

Such groups contain no elements that are more than exponentially distorted.

The paper discusses algebraic constraints and provides examples of these groups.

Abstract

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted.