= 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.
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