On maximal stable quotients of definable groups in NIP theories

TL;DR

This paper proves that for groups definable in NIP theories, there exists a minimal type-definable subgroup making the quotient stable, extending the concept of G^00 to a broader context.

Contribution

It introduces the concept of maximal stable quotients for definable groups in NIP theories, generalizing the G^00 subgroup.

Findings

Existence of a smallest type-definable subgroup with stable quotient

Extension of G^00 concept to broader class of groups

Provides a new structural understanding of definable groups in NIP theories

Abstract

For G a group definable in an NIP theory we prove that there is a smallest type-definable subgroup H of G such that the quotient G/H is stable. This generalizes the existence of G^00, the smallest type-definable subgroup of G of bounded index.