Shaun Allison : Classification by TSI Polish group actions

Posted by Filippo Calderoni , part of the Logic Seminar.

At

Sept. 1, 2020, 2 p.m.

In

Zoom

Abstract

Two dynamical conditions for orbit equivalence relations will be introduced -- the first providing an obstruction to classification by TSI Polish groups, and the second providing an obstruction to classification by non-Archimedean TSI Polish groups. A Polish group is TSI iff it has a compatible two-sided invariant metric. Following work of Hjorth and Drucker, we mimic the Scott analysis of countable structures to understand actions of general TSI Polish groups. Answering a question of Clemens and Coskey, these methods are used to show that the $\mathbb{Z}$-jump of $E_0$ is not classifiable by TSI Polish groups. Time permitting, we will show that if $E$ is Borel-reducible to $=^+$ and also classifiable by a non-Archimedean TSI Polish group, then it is Borel-reducible to $E_\infty^\omega$. Much of this work is joint with Aristotelis Panagiotopoulos.