= Cyclic reduction of a shortest conjugacy representative
A nonempty <shortest conjugacy representative> is a <cyclically reduced word>. If $w=s v s^{-1}$ with first and last letters inverse, $v$ is a shorter word representing a conjugate. Consequently $w^n$ has length exactly $n|w|$ as a <freely reduced word>, and any segment of the periodic word of length at most $|w|$ fits in a cyclic rotation of $w$. This observation turns a Dehn shortening segment crossing copy boundaries into a shortening of a conjugacy representative.
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