On minimal shapes and isoperimetric constants in hyperbolic lattices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913097636118528 |
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| author | D'Achille, Matteo Jacquier, Vanessa Ruszel, Wioletta M. |
| author_facet | D'Achille, Matteo Jacquier, Vanessa Ruszel, Wioletta M. |
| contents | We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in Häggström-Jonasson-Lyons. In fact, our balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On minimal shapes and isoperimetric constants in hyperbolic lattices D'Achille, Matteo Jacquier, Vanessa Ruszel, Wioletta M. Combinatorics Algebraic Topology Group Theory Number Theory Probability 05B45, 05C10, 05C69, 52B60, 11B68 We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in Häggström-Jonasson-Lyons. In fact, our balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices. |
| title | On minimal shapes and isoperimetric constants in hyperbolic lattices |
| topic | Combinatorics Algebraic Topology Group Theory Number Theory Probability 05B45, 05C10, 05C69, 52B60, 11B68 |
| url | https://arxiv.org/abs/2504.14080 |