Torsion Subgroups of Groups with Quadratic Dehn Function

We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are “optimal” in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 2 Superscript 48"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>48</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">n\geq 2^{48}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either odd or divisible by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript 9"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>9</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">2^9</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, any infinite free Burnside group with exponent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.