Scissors automorphism groups and their homology
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.
Cite
@article{arxiv.2408.08081,
title = {Scissors automorphism groups and their homology},
author = {Alexander Kupers and Ezekiel Lemann and Cary Malkiewich and Jeremy Miller and Robin J. Sroka},
journal= {arXiv preprint arXiv:2408.08081},
year = {2024}
}
Comments
58 pages, 16 figures. Comments welcome! v2: Corrected examples 6.35-37, updated references