#### Vol. 287, No. 1, 2017

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Groups of PL-homeomorphisms admitting nontrivial invariant characters

### Daciberg Lima Gonçalves, Parameswaran Sankaran and Ralph Strebel

Vol. 287 (2017), No. 1, 101–158
##### 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
##### Authors
 Daciberg Lima Gonçalves Instituto de Matemática e Estatística da Universidade de São Paulo Departamento de Matemática Rua do Matão, 1010 CEP 05508-090 São Paulo-SP Brazil Parameswaran Sankaran The Institute of Mathematical Sciences CIT Campus Chennai 600113 India Ralph Strebel Département de Mathématiques Université de Fribourg Chemin du Musée 23 CH-1700 Fribourg Switzerland