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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.15978 |
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| _version_ | 1866929688008458240 |
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| author | Delecroix, Vincent Legrand-Duchesne, Clément |
| author_facet | Delecroix, Vincent Legrand-Duchesne, Clément |
| contents | We consider a combinatorial reconfiguration problem on a subclass of quadrangulations of surfaces called square-tiled surfaces. Our elementary move is a shear in a cylinder that corresponds to a well-chosen sequence of diagonal flips that preserves the square-tiled properties. We conjecture that the connected components of this reconfiguration problem are in bijection with the connected components of the moduli space of quadratic differentials. We prove that the conjecture holds in the so-called hyperelliptic components of Abelian square-tiled surfaces. More precisely, we show that any two such square-tiled surfaces of genus $g$ can be connected by $O(g)$ powers of cylinder shears. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15978 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reconfiguration of square-tiled surfaces Delecroix, Vincent Legrand-Duchesne, Clément Combinatorics Algebraic Geometry Geometric Topology We consider a combinatorial reconfiguration problem on a subclass of quadrangulations of surfaces called square-tiled surfaces. Our elementary move is a shear in a cylinder that corresponds to a well-chosen sequence of diagonal flips that preserves the square-tiled properties. We conjecture that the connected components of this reconfiguration problem are in bijection with the connected components of the moduli space of quadratic differentials. We prove that the conjecture holds in the so-called hyperelliptic components of Abelian square-tiled surfaces. More precisely, we show that any two such square-tiled surfaces of genus $g$ can be connected by $O(g)$ powers of cylinder shears. |
| title | Reconfiguration of square-tiled surfaces |
| topic | Combinatorics Algebraic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2501.15978 |