Scissors automorphism groups I: Homological stability and K-theory

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. In the classical case of scissors congruence of polytopes, this leads to calculations and structural results on the homology of the scissors automorphism group. In two sequels, we show that more generally our approach leads to homology calculations for various families of groups appearing in group theory and dynamics, recovering results of Szymik-Wahl, Li, and Tanner.

v3: Accepted version, to appear in J. Reine Angew. Math. Changed title from "Scissors automorphism groups and their homology". Updated arguments to account for the role of Zylev's theorem, and revised according to referees' suggestions. Removed computational part of Sec. 5 and all of Sec. 6; these will reappear in the forthcoming paper "Scissors automorphism groups III: Homology calculations"