Torsion conjugacy bound for a Dehn presentation ($|w|\le \lfloor |r|/2\rfloor$)

In a finite Dehn presentation, every nonidentity finite-order element has a shortest conjugacy representative of length at most half the length of some relator. Indeed a positive power contains a Dehn segment. If the representative had length at least $k=\lfloor |r|/2\rfloor +1$, the first $k$ letters of that segment would fit into a cyclic rotation and could be replaced by fewer letters, contradicting minimality. Bounded relator lengths and a finite alphabet therefore give finitely many conjugacy classes of finite-order elements. With no relators the group is free and torsion-free, so only the identity class remains.