Cohomological and quasi-isometric diversity of groups with property $R_\infty$
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arXiv
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2026
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| author | Dekimpe, Karel de Araujo, Paula M. Lins Rego, Yuri Santos |
| author_facet | Dekimpe, Karel de Araujo, Paula M. Lins Rego, Yuri Santos |
| contents | How rich is the collection of groups with a given prominent property? In this work we approach this question for property~$R_\infty$, which says that every automorphism $φ$ of a given group has infinitely many orbits under the $φ$-twisted conjugation action $(g,x) \mapsto gxφ(g)^{-1}$. Generalising the soluble groups of Herbert Abels to a large family over many integral domains, we prove that most such groups have property~$R_\infty$ drawing from a classical result of Levchuk and a swift observation by Jabara. Within the broad programme of cataloguing finitely generated groups up to quasi-isometry, our groups can then be separated by finiteness properties and cohomological dimension whilst having~$R_\infty$. Abandoning finite presentability, we establish that property~$R_\infty$ is very abundant in a strong sense: there are uncountably many finitely generated groups (which can all be chosen to be amenable or non-amenable) that have~$R_\infty$ and are pairwise not quasi-isometric. The proofs vary in flavour. On the amenable side we use carefully constructed quotients of Abels' groups and a general strategy for quasi-isometric diversity established by Minasyan, Osin, and Witzel. For the non-amenable constructions we rely on modifications of Leary's type $\mathtt{FP}$ groups, further cohomological arguments, and recent powerful criteria for~$R_\infty$ due to Iveson, Martino, Sgobbi, Wong, and Fournier-Facio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17411 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cohomological and quasi-isometric diversity of groups with property $R_\infty$ Dekimpe, Karel de Araujo, Paula M. Lins Rego, Yuri Santos Group Theory Algebraic Topology 20E36, 20F16, 20F65, 20H25, 20J05, 57M07 How rich is the collection of groups with a given prominent property? In this work we approach this question for property~$R_\infty$, which says that every automorphism $φ$ of a given group has infinitely many orbits under the $φ$-twisted conjugation action $(g,x) \mapsto gxφ(g)^{-1}$. Generalising the soluble groups of Herbert Abels to a large family over many integral domains, we prove that most such groups have property~$R_\infty$ drawing from a classical result of Levchuk and a swift observation by Jabara. Within the broad programme of cataloguing finitely generated groups up to quasi-isometry, our groups can then be separated by finiteness properties and cohomological dimension whilst having~$R_\infty$. Abandoning finite presentability, we establish that property~$R_\infty$ is very abundant in a strong sense: there are uncountably many finitely generated groups (which can all be chosen to be amenable or non-amenable) that have~$R_\infty$ and are pairwise not quasi-isometric. The proofs vary in flavour. On the amenable side we use carefully constructed quotients of Abels' groups and a general strategy for quasi-isometric diversity established by Minasyan, Osin, and Witzel. For the non-amenable constructions we rely on modifications of Leary's type $\mathtt{FP}$ groups, further cohomological arguments, and recent powerful criteria for~$R_\infty$ due to Iveson, Martino, Sgobbi, Wong, and Fournier-Facio. |
| title | Cohomological and quasi-isometric diversity of groups with property $R_\infty$ |
| topic | Group Theory Algebraic Topology 20E36, 20F16, 20F65, 20H25, 20J05, 57M07 |
| url | https://arxiv.org/abs/2602.17411 |