Ulla Karhumäki: Small infinite simple groups of finite Morley rank with a tight automorphism whose fixed point subgroup is pseudofinite

Abstract

In the late 70’s, Gregory Cherlin and Boris I. Zilber independently conjectured that infinite simple groups of finite Morley rank are isomorphic to linear algebraic groups over algebraically closed fields.

We explain in detail a recent approach, developed by Pınar Uǧurlu, towards the Cherlin-Zilber conjecture. The main aim of this approach is to prove that the Cherlin-Zilber conjecture is equivalent to another conjecture called the Principal conjecture; Let \(G\) be an infinite simple group of finite Morley rank with a generic automorphism \(\alpha\) . Then the fixed point subgroup \(C_G(\alpha)\)  is pseudofinite. We prove a result supporting the expected equivalence between these two conjectures. Namely, we prove that, under suitable assumptions, a “small” infinite simple group of finite Morley rank \(G\)  admitting a tight automorphism \(\alpha\)  whose fixed point subgroup \(C_G(\alpha)\)  is pseudofinite is isomorphic to the Chevalley group \(PSL_2(K)\)  over an algebraically closed field \(K\)  of positive characteristic different from 2. This is joint work with Pınar Uǧurlu.

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