Tue, 8 Nov 2022, 1:25 pm MST
We discuss a geometry of involutions arising in "quasi-Frobenius" groups of finite Morley rank: groups possessing a quasi-self-normalizing subgroup $C$ whose conjugates intersect trivially. The resulting geometry is that of a generically defined projective 3-space, and our focus is on whether it is genuine or not: a distinction that separates $\operatorname{SO}_3(\mathbb{R})$ and $\operatorname{PGL}_2(\mathbb{C})$. We conjecture natural conditions on quasi-Frobenius groups of finite Morley rank to identify $\operatorname{PGL}_2(\mathbb{K})$, discuss progress on the conjecture (by us and, more recently, by Zamour), and highlight applications.
This is joint work with Adrien Deloro.