UIC Model Theory Seminar

Abstract:
The general goal is to extend some techniques from stable groups (e.g. forking calculus) to show structural results about groups which are not necessarily stable. In order to do that we need to assume that in our group there is a notion of independence relation satisfying some minimal list of nice properties. Such groups are called rosy groups. The coarsest relation with these nice properties is called thorn-independence.

We concentrate on structural results about superrosy groups with small thorn-U-rank (the rank defined by means of thorn-independence in the same way as Lascar U-rank is defined by means of forking independence), additionally assuming NIP (non independence property) and FSG (finitely satisfiable generics). In particular, our results generalize some theorems about superstable groups and definably connected, definably compact groups definable in o-minimal structures.

During my talk I will concentrate on the following result.
Theorem 1. Each group satisfying NIP, having FSG and of thorn-U-rank 1 is abelian-by-finite.

I will also discuss some partial results about the following conjecture.
Conjecture 2. Each group satisfying NIP, having hereditarily FSG and of thorn-U-rank 2 is solvable-by-finite.