Zoé Chatzidakis (École Normale Supérieure – CNRS)

Measures on perfect PAC fields

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This is work in progress, joint with Nick Ramsey (UCLA).

A conjecture, now disproved by Chernikov, Hrushovski, Kruckman, Krupinski, Pillay and Ramsey, asked whether any group with a simple theory is definably amenable.

It is well known that the counting measure on finite fields gives rise to a non-standard counting measure on pseudo-finite fields (the infinite models of the theory of finite fields). It was unknown whether other PAC fields possessed a reasonable measure, and in this talk, we will show that some of them do, although the measure we define does not have all the nice properties of a counting measure when the field is not pseudo-finite. This result can be used to show that if $G$ is a group definable in an $e$-free perfect PAC field, then $G$ is definably amenable. It extends to groups definable in $\omega$-free PAC fields. I will also discuss possible extensions to wider classes of perfect PAC fields.