Scissors automorphism groups and their homology

TL;DR

This paper explores the homology of scissors automorphism groups in categories with covers, linking it to scissors congruence K-theory and applying it to various complex groups to derive new and existing results.

Contribution

It establishes the independence of homology of scissors automorphism groups from objects and connects it to Zakharevich's scissors congruence K-theory spectrum, providing new computational methods.

Findings

Homology of scissors automorphism groups is object-independent under mild conditions.

Homology can be expressed via scissors congruence K-theory spectrum.

Results applied to interval exchange and Brin--Thompson groups, recovering and extending known results.

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.