Scissors automorphism groups and their homology

Alexander Kupers; Ezekiel Lemann; Cary Malkiewich; Jeremy Miller; Robin J. Sroka

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arXiv · 2408.08081

Scissors automorphism groups and their homology

Alexander Kupers·Ezekiel Lemann·Cary Malkiewich·Jeremy Miller·Robin J. Sroka

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Abstract

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.

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BibTeX RIS

Alexander Kupers, Ezekiel Lemann, Cary Malkiewich, Jeremy Miller, Robin J. Sroka. 2024-08-15. Scissors automorphism groups and their homology. https://arxiv.org/abs/2408.08081

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