Property $R_{\infty}$ for groups with infinitely many ends
Abstract
We show that an accessible group with infinitely many ends has property . That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property .
Cite
@article{arxiv.2504.12002,
title = {Property $R_{\infty}$ for groups with infinitely many ends},
author = {Francesco Fournier-Facio and Harry Iveson and Armando Martino and Wagner Sgobbi and Peter Wong},
journal= {arXiv preprint arXiv:2504.12002},
year = {2026}
}
Comments
28 pages. After submitting the first version on arXiv, we were contacted by Francesco Fournier-Facio who observed that the results could also be obtained via the use of quasimorphisms. He has written us an appendix explaining this and extending the results. Since then, we have decided to include Francesco Fournier-Facio as a full co-author