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Scissors automorphism groups and their homology

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arxiv 2408.08081 v2 pith:OGETZIA3 submitted 2024-08-15 math.KT math.ATmath.GR

Alexander Kupers , Ezekiel Lemann , Cary Malkiewich , Jeremy Miller , Robin J. Sroka This is my paper · ORCID
classification math.KTmath.ATmath.GR
keywords groupsscissorshomologyautomorphismcongruencegroupk-theoryobject
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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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Dennis Trace for Assembler K-Theory

    math.AT 2026-07 conditional novelty 7.5 of 10

    A Dennis trace from assembler K-theory to Hochschild homology of scissors correspondences refines the regulator, making group homology a trace invariant.

  2. Piecewise isometry groups of Euclidean tessellations

    math.GR 2026-07 conditional novelty 7.0 of 10

    Every piecewise isometry group of a cocompact Euclidean tessellation by finitely many hyperplane families is elementary amenable.

  3. Representation stability for ordered Hurwitz spaces

    math.AT 2025-09 conditional novelty 7.0 of 10

    The homology groups of ordered Hurwitz spaces, seen as representations of symmetric groups, have stable multiplicities in a range that grows linearly with homological degree.

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