Torsion, torsion length and finitely presented groups

Abstract

Abstract We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some C ′ ⁢ ( 1 / 6 ) {C^{\prime}(1/6)} finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is C ′ ⁢ ( 1 / 6 ) {C^{\prime}(1/6)} , and thus word-hyperbolic and virtually torsion-free.