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.