Property R∞ for groups with infinitely many ends

Fournier-Facio, Francesco, Iveson, Harry Margaret John, Martino, Armando, Sgobbi, Wagner and Wong, Peter (2026) Property R∞ for groups with infinitely many ends. Geometriae Dedicata, 220, [25]. (doi:10.1007/s10711-026-01078-x).

Record type: Article

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∞.

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More information

Accepted/In Press date: 26 February 2026

e-pub ahead of print date: 19 March 2026

Published date: 19 March 2026

Additional Information: 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

Keywords: math.GR, math.GT

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