Scissors automorphism groups and their homology
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arXiv
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| Format: | Preprint |
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2024
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| author | Kupers, Alexander Lemann, Ezekiel Malkiewich, Cary Miller, Jeremy Sroka, Robin J. |
| author_facet | Kupers, Alexander Lemann, Ezekiel Malkiewich, Cary Miller, Jeremy Sroka, Robin J. |
| contents | In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. We apply this to various families of groups, such as interval exchange groups and Brin--Thompson groups, recovering results of Szymik--Wahl, Li, and Tanner, and obtaining new results as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scissors automorphism groups and their homology Kupers, Alexander Lemann, Ezekiel Malkiewich, Cary Miller, Jeremy Sroka, Robin J. K-Theory and Homology Algebraic Topology Group Theory 18F25, 19D99, 20J05, 55P42, 52B45 In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. We apply this to various families of groups, such as interval exchange groups and Brin--Thompson groups, recovering results of Szymik--Wahl, Li, and Tanner, and obtaining new results as well. |
| title | Scissors automorphism groups and their homology |
| topic | K-Theory and Homology Algebraic Topology Group Theory 18F25, 19D99, 20J05, 55P42, 52B45 |
| url | https://arxiv.org/abs/2408.08081 |