Abstract

Two polytopes in Euclidean n-space are called scissors congruent if one can be cut into finitely many polytopic pieces that can be rearranged by Euclidean isometries to form the other. A generalized version of Hilbert’s third problem asks for a classification of Euclidean n-polytopes up to scissors congruence. In this talk, we consider the complementary question and study the scissors automorphism group—it encodes all transformations realizing the scissors congruence relation between distinct polytopes. This leads to a group-theoretic interpretation of Zakharevich’s higher scissors congruence K-theory. By varying the notion of polytope, scissors automorphism groups recover many important examples of groups appearing in dynamics and geometric group theory including Brin—Thompson groups and groups of rectangular exchange transformations. Combined with recently developed computational tools for scissors congruence K-theory, we recover and extend calculations of their homology. This talk is based on joint work with Kupers—Lemann—Malkiewich—Miller.