Polynomial hyperbolicity and products of free groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917514290659328 |
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| author | Genevois, Anthony |
| author_facet | Genevois, Anthony |
| contents | In this article, we define a locally finite graph $X$ as $η$-polynomially hyperbolic if there exists a Lipschitz map $φ: X \to Z$ to some hyperbolic space $Z$ satisfying the following condition: there exists $C \geq 0$ such that $$|B(p,R_1) \cap φ^{-1} (B(q,R_2))| \leq (C R_1)^{η(C R_2)} \text{ for all } p,q \in X, R_1,R_2 \geq 0.$$ The picture to keep in mind is that coarse fibres of $φ$ have polynomial growth with a degree coarsely controlled by $η$ as the thickness of the fibres grows. The map $η$ quantifies how brutal we have to be in order to turn $X$ into a hyperbolic space. Our main result is that, among cocompact special groups, being $\mathrm{lin}$-polynomially hyperbolic amounts not to contain $\mathbb{F}_2 \times \mathbb{F}_2$ as a subgroup. Consequently, containing $\mathbb{F}_2 \times \mathbb{F}_2$ as a subgroup turns out to be quasi-isometric invariant for cocompact special groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_20419 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial hyperbolicity and products of free groups Genevois, Anthony Group Theory Metric Geometry 20F65, 20F69, 20F67 In this article, we define a locally finite graph $X$ as $η$-polynomially hyperbolic if there exists a Lipschitz map $φ: X \to Z$ to some hyperbolic space $Z$ satisfying the following condition: there exists $C \geq 0$ such that $$|B(p,R_1) \cap φ^{-1} (B(q,R_2))| \leq (C R_1)^{η(C R_2)} \text{ for all } p,q \in X, R_1,R_2 \geq 0.$$ The picture to keep in mind is that coarse fibres of $φ$ have polynomial growth with a degree coarsely controlled by $η$ as the thickness of the fibres grows. The map $η$ quantifies how brutal we have to be in order to turn $X$ into a hyperbolic space. Our main result is that, among cocompact special groups, being $\mathrm{lin}$-polynomially hyperbolic amounts not to contain $\mathbb{F}_2 \times \mathbb{F}_2$ as a subgroup. Consequently, containing $\mathbb{F}_2 \times \mathbb{F}_2$ as a subgroup turns out to be quasi-isometric invariant for cocompact special groups. |
| title | Polynomial hyperbolicity and products of free groups |
| topic | Group Theory Metric Geometry 20F65, 20F69, 20F67 |
| url | https://arxiv.org/abs/2605.20419 |