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Logic Seminar

Wednesday, November 10, 2021
12:00pm to 1:00pm
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Torsion-free abelian groups are Borel complete
Gianluca Paolini, Department of Mathematics, University of Torino,

I will talk about my recent result joint with S. Shelah establishing that the Borel space of torsion-free abelian groups with domain ω is Borel complete, i.e., that the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989. Time permitting, I will also talk about some recent results (also joint with S. Shelah) on the existence of uncountable Hopfian and co-Hopfian abelian groups and on anti-classification results for the countable co-Hopfian abelian and 2-nilpotent groups. In particular, we will see that the countable co-Hopfian groups are complete co-analytic in the Borel space of 2-nilpotent groups with domain ω. This solves an open question of Thomas, who had posed the question for the space of all groups with domain ω.

For more information, please email A. Kechris at [email protected].