Sigma 1 of PL-homeomorphism groups

TL;DR

This paper surveys known results and introduces new findings on the Sigma 1 invariant of PL-homeomorphism groups acting on intervals, lines, and half-lines, using Cayley-graph methods.

Contribution

It provides a comprehensive survey and new results on the Sigma 1 invariant for PL-homeomorphism groups, expanding understanding of their algebraic properties.

Findings

New results on Sigma 1 for groups of PL-homeomorphisms of lines and half-lines.

A unified approach using Cayley-graph definitions.

Enhanced understanding of the invariant's behavior in different settings.

Abstract

I survey the known results about the invariant Sigma 1 of groups of PL-homeomorphisms of a compact interval and supplement them with new results about Sigma 1 of PL-homeomorphism groups of a half line or a line. The proofs are based on the Cayley-graph definition of the invariant.