Gromov's Vision: The Phase Transition Between Amenability and Hyperbolicity
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| Format: | Recurso digital |
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2026
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| _version_ | 1866901910460563456 |
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | Mikhail Gromov's profound contributions to geometric group theory fundamentally reshaped the study of infinite discrete groups by embedding their algebraic properties within a geometric framework. This paper explores Gromov's seminal concept of a "phase transition" between amenability and hyperbolicity, two diametrically opposed large-scale geometric properties of groups. Amenable groups are characterized by their "non-paradoxical" nature and often exhibit sub-exponential or polynomial growth, while hyperbolic groups, introduced by Gromov, possess a strong negative curvature in their Cayley graphs and typically display exponential growth. The notion of a phase transition, borrowed from physics, serves as a powerful metaphor to describe the sharp demarcation line observed in the behavior of groups as their geometric properties shift. This paper delves into the definitions, key properties, and the interplay between amenability and hyperbolicity, emphasizing how Gromov's work illuminates this transition. We discuss the geometric invariants, such as quasi-isometry, that are crucial in understanding this large-scale geometry. Furthermore, we examine groups that lie at or near this theoretical boundary, illustrating the delicate balance of properties that define the transition. The discussion highlights the theoretical implications of this dichotomy for group structure, algorithmic problems, and broader connections within mathematics, affirming Gromov's vision as a cornerstone of modern geometric group theory. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18724378 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Gromov's Vision: The Phase Transition Between Amenability and Hyperbolicity Revista, Zen MATH, 10 Mikhail Gromov's profound contributions to geometric group theory fundamentally reshaped the study of infinite discrete groups by embedding their algebraic properties within a geometric framework. This paper explores Gromov's seminal concept of a "phase transition" between amenability and hyperbolicity, two diametrically opposed large-scale geometric properties of groups. Amenable groups are characterized by their "non-paradoxical" nature and often exhibit sub-exponential or polynomial growth, while hyperbolic groups, introduced by Gromov, possess a strong negative curvature in their Cayley graphs and typically display exponential growth. The notion of a phase transition, borrowed from physics, serves as a powerful metaphor to describe the sharp demarcation line observed in the behavior of groups as their geometric properties shift. This paper delves into the definitions, key properties, and the interplay between amenability and hyperbolicity, emphasizing how Gromov's work illuminates this transition. We discuss the geometric invariants, such as quasi-isometry, that are crucial in understanding this large-scale geometry. Furthermore, we examine groups that lie at or near this theoretical boundary, illustrating the delicate balance of properties that define the transition. The discussion highlights the theoretical implications of this dichotomy for group structure, algorithmic problems, and broader connections within mathematics, affirming Gromov's vision as a cornerstone of modern geometric group theory. |
| title | Gromov's Vision: The Phase Transition Between Amenability and Hyperbolicity |
| url | https://doi.org/10.5281/zenodo.18724378 |