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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.11991 |
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| _version_ | 1866915945339944960 |
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| author | Duda, Karol |
| author_facet | Duda, Karol |
| contents | We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship between C(6) complexes and their duals. In the C(4)-T(4) case we demonstrate that genuine flats do not necessarily exist, providing an explicit example of a C(4)-T(4) complex with an action of $\mathbb{Z}^2$ without invariant flat, and hence not admitting any CAT(0) metric invariant under automorpihsms. We introduce the notion of thickened-flats and prove a Flat Torus Theorem for quasi-flats by passing to quadric complexes via quadrization and invoking the Quadric Flat Torus Theorem of Hoda-Munro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The small cancellation flat torus theorem Duda, Karol Group Theory We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship between C(6) complexes and their duals. In the C(4)-T(4) case we demonstrate that genuine flats do not necessarily exist, providing an explicit example of a C(4)-T(4) complex with an action of $\mathbb{Z}^2$ without invariant flat, and hence not admitting any CAT(0) metric invariant under automorpihsms. We introduce the notion of thickened-flats and prove a Flat Torus Theorem for quasi-flats by passing to quadric complexes via quadrization and invoking the Quadric Flat Torus Theorem of Hoda-Munro. |
| title | The small cancellation flat torus theorem |
| topic | Group Theory |
| url | https://arxiv.org/abs/2601.11991 |