We discuss a geometry of involutions arising in "quasi-Frobenius" groups of finite Morley rank: groups possessing a quasi-self-normalizing subgroup H 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 SO3(R) and PGL2(C). We conjecture natural conditions on quasi-Frobenius groups of finite Morley rank to identify PGL2(K), discuss progress on the conjecture (by us and, more recently, by Zamour), and highlight applications.
Geometry of involutions in ranked groups