Resumen

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.

A homeomorphism of the Cantor set is contained in the topological full group of a Cantor dynamical system if the Cantor set can be cut into finitely many clopen pieces on which the homeomorphism is described by the dynamics. Conjectures of Matui predict a relation between this group, an ample groupoid encoding the dynamics and the K-theory of a related C*-algebra.

The goal of this talk is to outline analogies between these two subjects. We will consider the scissors automorphism group -- it encodes all transformations realizing the scissors congruence relation between distinct polytopes and can be seen as an analogue of the topological full group in dynamics. This leads to a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, and relates Malkiewich's calculation of these K-groups for Euclidean 1-space to recent advances of Li and Tanner on Matui's conjectures. Using computational tools for scissors congruence K-theory, we recover and extend group homology calculations for Brin--Thompson groups and groups of rectangular exchange transformations. All of this is based on joint work with Kupers--Lemann--Malkiewich--Miller.

Este seminario, abierto a todo el público interesado, es organizado por los miembros del grupo de Geometría No Conmutativa. Las reuniones son los martes a las 10:00hs, vía Zoom (para recibir las coordenadas escribir a gtartaglia arroba mate.unlp.edu.ar).