Groups of PL-homeomorphisms admitting non-trivial invariant characters

TL;DR

This paper investigates certain groups of piecewise linear homeomorphisms of the real line, showing they admit special invariant characters and have infinitely many twisted conjugacy classes, extending known results about Thompson-like groups.

Contribution

It demonstrates that various classes of PL-homeomorphism groups possess non-trivial automorphism-invariant characters and infinite twisted conjugacy classes, generalizing previous findings.

Findings

Existence of non-trivial automorphism-invariant characters in these groups

Groups have infinitely many twisted conjugacy classes

Extension of known properties of Thompson's group F to broader classes

Abstract

We show that several classes of groups G of PL-homeomorphisms of the real line admit non-trivial homomorphisms from G to the additive group of reals that are fixed by every automorphism of G. The classes of groups enjoying the stated property include the generalisations of Thompson's group F studied by K. S. Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.