Looking for stabilizers in NSOP$\_1$

TL;DR

This paper explores the properties of groups definable in NSOP1 theories, introducing the Kim-stabilizer concept and demonstrating its applications to algebraic groups over certain fields, revealing new structural insights.

Contribution

It introduces the Kim-stabilizer in NSOP1 theories and applies it to relate definable groups to algebraic groups over specific fields, expanding understanding of group behavior in these theories.

Findings

Existence of Kim-stabilizers in NSOP1 groups.

Finite-to-one embeddings into algebraic groups.

Definable groups in omega-free PAC fields satisfy key conditions.

Abstract

In this work we study some examples of groups definable and type-definable in NSOP1 theories. We exhibit some behaviors of these groups that differ from the ones of simple groups. We take interest in the notions of generics and stabilizers, and define the Kim-stabilizer. We apply the notion of Kim-stabilizer and the stabilizer from Hrushovsky to the context of a group G definable in an NSOP1 field F satisfying some assumptions to show that there is a finite to one embedding of a type definable subgroup of G of bounded index into an algebraic group over F. We then show that definable groups in omega-free PAC fields satisfy these conditions.