Superrosy dependent groups having finitely satisfiable generics

TL;DR

This paper investigates the structure of superrosy dependent groups with finitely satisfiable generics, establishing conditions under which such groups are abelian-by-finite or solvable-by-finite, and showing fields are algebraically closed.

Contribution

It generalizes known results from finite Morley rank and o-minimal structures to superrosy dependent groups with finitely satisfiable generics, providing new structural classifications.

Findings

Thorn rank 1 groups are abelian-by-finite.

Thorn rank 2 groups are solvable-by-finite.

Fields in this context are algebraically closed.

Abstract

We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming thorn rank 1, the group is abelian-by-finite, and assuming thorn rank 2 the group is solvable by finite. Also a field is algebraically closed.