= Solution
Minimality of the conjugacy representative implies that $w$ is freely reduced; otherwise <free reduction> produces a shorter representative of the same element. It is also a <cyclically reduced word>. If $w=svs^{-1}$ as a <freely reduced word> with its first and last letters inverse, the shorter word $v$ represents a conjugate, contradicting minimality. This is the <cyclic reduction of a shortest conjugacy representative>.
The word is nonempty because the element has order greater than one. Since it is cyclically reduced, no cancellation occurs between successive copies in $w^n$, and $w^n$ is a nonempty <freely reduced word>. It represents the identity because <conjugation> preserves the order. Apply the defining property of the <Dehn presentation> to obtain
$$
\boxed{u\text{ is a consecutive segment of }w^n\text{ and of some }r\in R,
\qquad |u|>|r|/2.}
$$
The <relator> belongs to the finite symmetrized set fixed in the definition. No assumption that the group itself is torsion-free is made.
Back to article page
