Gromov Hyperbolicity: A Sharp Barrier to Amenability in Infinite Groups
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2026
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper explores the intricate relationship between Gromov hyperbolicity and amenability within the realm of infinite discrete groups. Gromov hyperbolicity, a geometric property akin to negative curvature, imposes strong structural constraints on groups, dictating their large-scale geometry. Amenability, an analytic property defined by the existence of an invariant mean on the group, signifies a certain "niceness" in terms of averaging. We argue that for infinite groups, particularly non-elementary ones, Gromov hyperbolicity acts as a sharp barrier to amenability. While elementary hyperbolic groups, such as virtually cyclic groups, are amenable, the vast majority of non-elementary hyperbolic groups are non-amenable, often containing non-abelian free subgroups. This fundamental dichotomy arises from the inherent exponential growth and expansion properties characteristic of non-elementary hyperbolic groups, which directly contradict the F{o}lner conditions central to amenability. The paper reviews the theoretical underpinnings of both concepts, synthesizes key results demonstrating their antagonism, and discusses the profound implications for the classification and understanding of infinite groups in geometric group theory and related fields. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18140437 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Gromov Hyperbolicity: A Sharp Barrier to Amenability in Infinite Groups Revista, Zen MATH, 10 This paper explores the intricate relationship between Gromov hyperbolicity and amenability within the realm of infinite discrete groups. Gromov hyperbolicity, a geometric property akin to negative curvature, imposes strong structural constraints on groups, dictating their large-scale geometry. Amenability, an analytic property defined by the existence of an invariant mean on the group, signifies a certain "niceness" in terms of averaging. We argue that for infinite groups, particularly non-elementary ones, Gromov hyperbolicity acts as a sharp barrier to amenability. While elementary hyperbolic groups, such as virtually cyclic groups, are amenable, the vast majority of non-elementary hyperbolic groups are non-amenable, often containing non-abelian free subgroups. This fundamental dichotomy arises from the inherent exponential growth and expansion properties characteristic of non-elementary hyperbolic groups, which directly contradict the F{o}lner conditions central to amenability. The paper reviews the theoretical underpinnings of both concepts, synthesizes key results demonstrating their antagonism, and discusses the profound implications for the classification and understanding of infinite groups in geometric group theory and related fields. |
| title | Gromov Hyperbolicity: A Sharp Barrier to Amenability in Infinite Groups |
| url | https://doi.org/10.5281/zenodo.18140437 |