Groups of PL-homeomorphisms admitting nontrivial invariant characters

Daciberg Lima Gonçalves, Parameswaran Sankaran and Ralph Strebel

Vol. 287 (2017), No. 1, 101–158

DOI: 10.2140/pjm.2017.287.101

Abstract

We show that several classes of groups $G$ of PL-homeomorphisms of the real line admit nontrivial homomorphisms $\chi :G\to ℝ$ that are fixed by every automorphism of $G$. The classes enjoying the stated property include the generalizations of Thompson’s group $F$ studied by K. S. Brown (1992), M. Stein (1992), S. Cleary (1995), and Bieri and Strebel (2016), but also the class of groups investigated by Bieri, Neumann, and Strebel (Theorem 8.1 in Invent. Math. 90 (1987), 451–477). It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.

Keywords

groups of PL-homeomorphisms of the real line, Bieri–Neumann–Strebel invariants, twisted conjugacy

Mathematical Subject Classification 2010

Primary: 20E45

Secondary: 20E36, 20F28

Milestones

Received: 12 December 2015

Revised: 28 June 2016

Accepted: 10 August 2016

Published: 6 February 2017