Groups and Dynamics Seminar

Date: June 3, 2020

Time: 12:00PM - 1:00PM

Location: 940 9667 3668

Speaker: Andrew Marks, UCLA

Title: Measurable realizations of abstract systems of congruence

Abstract: In the last few years, several new results have been proved with the unifying theme that the "paradoxical" sets in many classical geometrical paradoxes can surprisingly be much "nicer" than one would naively expect. For example, by the Banach-Tarski paradox any two bounded subsets A and B of R^3 with nonempty interior are equidecomposable. However, if A and B have the same Lebesgue measure, then a recent theorem of Grabowski, M\'ath\'e and Pikhurko states that A and B are equidecomposable using Lebesgue measurable pieces. So for example, there is a Lebesgue measurable equidecomposition of a cube and a ball in R^3 of the same volume An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and n-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the 2-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of PSL_2(Z) on P^1(R). This is joint work with Clinton Conley and Spencer Unger.