Mathematicians discovered that the action of certain large groups can lead to paradoxical decompositions. The Banach-Tarski paradox is one such, and can be attributed to the fact that the isometry group of 3-dimensional space contains a free group. Groups admitting paradoxical decompositions are called non-amenable. The von Neumann-Day problems asked whether a non-amenable (countable) group necessarily contains a free group. If we looked only inside the world of groups defined by real matrices then we would say yes. However the world of all countably infinite groups is vast and indeed one can find groups for which the answer is no. We will explore the concepts and resolution of the problem, on the way encountering groups which defy our intuition.
Pre-requisites: Algebra 2.
Useful to have: Geometry 3.
References:
Office hours with a geometric group theorist, M. Clay and D. Margalit.
Geometric group theory, C. Druţu, M. Kapovich
Amenability of discrete groups by examples, K. Juschenko