The Bender method in groups of finite Morley rank

TL;DR

This paper explores the structure of Borel subgroups in groups of finite Morley rank, using 0-unipotence theory to analyze intersections and establish abelian properties in non-tame minimal simple groups.

Contribution

It introduces a new toolkit based on 0-unipotence theory for analyzing Borel subgroup intersections in non-tame minimal simple groups.

Findings

Connected nilpotent subgroups in intersections are abelian

Jaligot's Lemma extends to non-tame groups with new methods

Provides insights into the structure of finite Morley rank groups

Abstract

Jaligot's Lemma states that the Fitting subgroups of distinct Borel subgroups do not intersect in a tame minimal simple groups of finite Morley. Such a strong result appears hopeless without tameness. Here we use the 0-unipotence theory to build a toolkit for the analysis of nonabelian intersections of Borel subgroups. As a demonstration, we show that any connected nilpotent subgroup of an intersection of Borel subgroups, in a nontame minimal simple group, must actually be abelian.