Canonical Reduction Systems in Artin-Tits groups of spherical type
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908580295213056 |
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| author | Cumplido, María González-Meneses, Juan Perego, Davide |
| author_facet | Cumplido, María González-Meneses, Juan Perego, Davide |
| contents | We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical Reduction Systems in Artin-Tits groups of spherical type Cumplido, María González-Meneses, Juan Perego, Davide Group Theory Geometric Topology 20F36, 57K20 We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period. |
| title | Canonical Reduction Systems in Artin-Tits groups of spherical type |
| topic | Group Theory Geometric Topology 20F36, 57K20 |
| url | https://arxiv.org/abs/2510.00713 |