Cherlin–Zilber Algebraicity Conjecture

Establish that every infinite simple group of finite Morley rank is isomorphic to a simple algebraic group over an algebraically closed field.

Background

Groups of finite Morley rank arise naturally in model theory as binding groups and admit a well-behaved notion of dimension (Morley rank). A central goal in the area is to show that such groups coincide with classical algebraic groups when they are simple, thereby bridging model-theoretic and algebraic structures.

This conjecture has been a driving force in the field since the pioneering work of Zilber and Cherlin and underpins many classification programs for groups of finite Morley rank.

References

Conjecture 1. The Cherlin-Zilber Algebraicity Conjecture. Infinite simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields.

— Primitive permutation groups of finite Morley rank and affine type  (2405.07307 - Berkman et al., 2024) in Section 1.4

While the conjecture remains open, the past two decades have seen a number of fundamental results that provide strong evidence for its validity in the presence of involutions.

— The Borovik-Cherlin conjecture holds in ACF  (2609.02198 - Karhumäki et al., 2 Sep 2026) in Section 1, Introduction