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Logic Seminar

Tuesday, January 31, 2023
12:00pm to 1:00pm
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Classification strength of Polish groups and involving S∞
Shaun Allison, Mathematical and Computational Sciences, University of Toronto,

Please note that the time is PST

We say that a Polish group G has stronger classification strength than a Polish group H iff every orbit equivalence relation induced by a continuous action of H on a Polish space is Borel-reducible to an orbit equivalence relation induced by a continuous action of G on a Polish space (or in other words, "classified" by G. The notion of classification strength gives a way to measure the inherent complexity of a Polish group, and gives rise to interesting hierarchies.
We say that G involves H iff there is a closed subgroup G′ of G and a continuous surjective homomorphism from G′ onto H. If G involves H then it has greater classification strength, a result of Mackey and Hjorth. Also, a result of Hjorth implies that any Polish group which has greater classification strength than S∞, the Polish group of permutations of a countably-infinite set, involves S∞. In other words, the non-Archimedean Polish groups (i.e. the closed subgroups of S∞) with maximal classification strength are exactly those which involve S∞.
We will describe several new necessary and sufficient conditions for a non-Archimedean Polish group to involve S∞, some of which may have independent interest in model theory. One result of particular significance is that if G classifies =+, a natural equivalence relation very low in the Borel hierarchy, then G must involve S∞. Moreover, a natural rank function of model-theoretic flavor arises, measuring how close a non-Archimedean Polish group is from involving S∞, which yields an interesting hierarchy of classification strength. I will also mention previous and ongoing work with Aristotelis Panagiotopoulos relating to another hierarchy of classification strength among the cli (complete left-invariant) Polish groups.

For more information, please contact Math Dept. by phone at 626-395-4335 or by email at A. Kechris at [email protected].