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Classification of generically 2-transitive linear actions

Classify connected groups G of finite Morley rank with finite center and simple quotient G/Z(G) that act definably and faithfully on an abelian group V, are transitive on V \ {0}, and generically 2-transitive, by proving that the action is definably equivalent to the natural action of SL_n(K) on K^n for some algebraically closed field K.

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Background

This conjecture seeks a linear classification of highly transitive actions with strong homogeneity on the nonzero elements of the abelian group, paralleling known results for algebraic groups in characteristic 0 and positive characteristic.

The algebraic-case analogue is known (Knop’s result), and this conjecture extends the expectation to groups of finite Morley rank.

References

Conjecture 6. [17, Conjecture 5] Let G be a connected group of finite Morley rank with Z(G) finite and G/Z(G) simple. Assume that G acts definably and faithfully on an abelian group V , and that G is

transitive on V \ {0} and generically 2-transitive. Then this action is definably equivalent to the natural action of (SL nK) on K ) for some algebraically closed field K.

— Primitive permutation groups of finite Morley rank and affine type (2405.07307 - Berkman et al., 12 May 2024) in Section 1.13