Property R∞ for groups with infinitely many ends

Abstract

We show that an accessible group with infinitely many ends has property R∞. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property R∞ 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 R∞.

In an appendix, Francesco Fournier-Facio shows that a group with a non-zero Aut-invariant homogeneous quasimorphism has property R∞, which applies to many groups with hyperbolic features.

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ORCID for Armando Martino: orcid.org/0000-0002-5350-3029

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