Groups and Dynamics Seminar

We give a short introduction to Borel complexity and how it formalizes the problem of determining whether two objects are equivalent. This theory can be used to yield anti-classification results, such as the non-existence of concrete invariants. We give a new dynamical criterion for complexity based on double cosets in locally compact second countable groups and use it to answer a question of Calderoni, Marker, Motto-Ros and Shani, as well as strengthen a recent result of Gao, Li and Sun. For example, we show that the action of SL_n(Z) acting on R^n is stably orbit equivalent to a free and pmp action of SL_{n-1}(R) ⋉ R^{n-1}.