Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank

TL;DR

This paper introduces mock hyperbolic reflection spaces to analyze Frobenius groups of finite Morley rank, revealing structural properties and rank constraints, especially for groups of odd type and low Morley rank.

Contribution

It defines mock hyperbolic reflection spaces and applies them to study Frobenius groups, establishing new rank inequalities and classification results in finite Morley rank groups.

Findings

Connected Frobenius groups of odd type and Morley rank ≤10 either split or are sharply 2-transitive.

Mock hyperbolic reflection spaces satisfy specific rank inequalities.

Groups of Morley rank 8 or 10 are characterized as simple non-split sharply 2-transitive groups.

Abstract

We define the notion of mock hyperbolic reflection spaces and use it to study Frobenius groups, in particular in the context of groups of finite Morley rank including the so-called bad groups. We show that connected Frobenius groups of finite Morley rank and odd type with nilpotent complement split or interpret a bad field of characteristic zero. Furthermore, we show that mock hyperbolic reflection spaces of finite Morley rank satisfy certain rank inequalities, implying in particular that any connected Frobenius group of odd type and Morley rank at most ten either splits or is a simple non-split sharply 2-transitive group of characteristic different from 2 and of Morley rank 8 or 10.