Omega-categorical groups and rings of finite dimension

TL;DR

This work studies $\omega$-categorical$ groups and rings of finite dimension, addressing structural classifications under finite dimensionality. It develops a framework of finite-dimensional theories, dimensional-generics, bilinear quasi-forms, and principal indiscernible sequences, culminating in a key induction on reduced dimension to prove virtual almost triviality of bilinear forms. The main results show that finite-dimensional $\omega$-categorical groups are $finite$-by-abelian-by-finite and finite-dimensional $\omega$-categorical rings are virtually finite-by-null, advancing the understanding of tame model-theoretic behavior in this setting. Overall, the paper extends meta-conjectures about tameness and nilpotency to finite-dimensional $\omega$-categorical structures, providing a decisive structural classification in this regime.

Abstract

We prove that a finite-dimensional omega-categorical group is finite-by-abelian-by-finite and that a finite-dimensional omega-categorical ring is virtually finite-by-null.

Theorems & Definitions (71)

• Definition
• Theorem 1
• Corollary 2
• proof
• Lemma 3
• Lemma 4
• Theorem 6
• Theorem 7
• Theorem 8
• Definition