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Generic expansion of an abelian variety by a subgroup

Let A be an abelian variety in an algebraically closed field of characteristic 0. We prove that the expansion of A by a generic divisible subgroup of A with the same torsion exists provided A has few algebraic endomorphisms, namely End(A)=Z . The resulting theory is NSOP1 and not simple. Note that there exist abelian varieties A with End(A)=Z of any genus.