Measures on bounded perfect PAC fields
Zoé Chatzidakis, Nicholas Ramsey
TL;DR
The paper constructs translation-invariant Keisler measures on definable sets in bounded perfect PAC fields, extending to perfect Frobenius fields, and uses a Markov-chain approach to build measures in a special case where the absolute Galois group is the universal Frattini cover of a finite group. By ultralimit arguments, these measures extend to all bounded perfect PAC fields and perfect Frobenius fields, enabling definable amenability of all definable groups in these settings via the Hrushovski–Pillay classification. The work connects combinatorial Markov-chain techniques with inverse-system model theory of absolute Galois groups to produce a robust toolset for analyzing definable groups in NSOP1/simple theories. It also explores the G-action regime through G-TCF and raises questions about measure values and the full scope of definable amenability across broader classes of fields.
Abstract
We describe a construction for producing Keisler measures on bounded perfect PAC fields. As a corollary, we deduce that all groups definable in bounded perfect PAC fields, and even in unbounded perfect Frobenius fields, are definably amenable. This work builds on our earlier constructions of measures for $e$-free PAC fields and a related construction due to Will Johnson.