jats:titleAbstract</jats:title> jats:pWe 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 <jats:inline-formula id="j_jgth-2018-0022_ineq_9999_w2aab3b7e4488b1b6b1aab1c15b1b1Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>6</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0022_eq_0139.png" /> jats:tex-math{C^{\prime}(1/6)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula id="j_jgth-2018-0022_ineq_9998_w2aab3b7e4488b1b6b1aab1c15b1b3Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>6</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0022_eq_0139.png" /> jats:tex-math{C^{\prime}(1/6)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and thus word-hyperbolic and virtually torsion-free.</jats:p>