# The geometry of involutions in ranked groups with a TI-subgroup

## TL;DR

This paper explores the geometric structure of involutions in groups of finite Morley rank, unifying previous results and proposing a new approach that could lead to an identification theorem for PGL_2(K).

## Contribution

It introduces a unified geometric framework for involutions in finite Morley rank groups and suggests a novel path toward an identification theorem for PGL_2(K).

## Key findings

- Unified the geometric analysis of involutions in finite Morley rank groups.
- Generalized and extended previous results in the area.
- Proposed a conjecture linking this geometry to an identification theorem.

## Abstract

We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this path leads to a new identification theorem for $\operatorname{PGL}_2(\mathbb{K})$.