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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.08081
- arXiv:
- arXiv:2408.08081
- Bibcode:
- 2024arXiv240808081K
- Keywords:
-
- Mathematics - K-Theory and Homology;
- Mathematics - Algebraic Topology;
- Mathematics - Group Theory;
- 18F25;
- 19D99;
- 20J05;
- 55P42;
- 52B45
- E-Print:
- 58 pages, 16 figures. Comments welcome! v2: Corrected examples 6.35-37, updated references