 Torsion in an HNN extension

Every finite-order element of an HNN extension is conjugate to a finite-order element of its base. Choose a conjugate minimizing the number of stable letters. If this number is positive, no pinch is possible either internally or across the end-to-start join, since a boundary pinch would allow a conjugation shortening it. Every positive power is then reduced with positive stable-letter length, and Britton's lemma makes it nonidentity. In particular, an HNN extension of a torsion-free group is torsion-free.