Abelian varieties over large algebraic fields with infinite torsion

Let A be a non-zero abelian variety defined over a number field K and let \(\overline K \) be a fixed algebraic closure of K. For each element σ of the absolute Galois group \({\text{Gal}}(\overline K /K)\), let \(\overline K (\sigma )\) be the fixed field in \(\overline K \) of σ. We show that the torsion subgroup of \(A(\overline K (\sigma ))\) is infinite for all \(\sigma \in {\text{Gal}}(\overline K /K)\) outside of some set of Haar measure zero. This proves the number field case of a conjecture of W.-D. Geyer and M. Jarden.