An Explicit Bound for the Conjugator Length Function of a Surface Group

For a surface group ${\pi }_1({\mathrm{\Sigma }}_g)=⟨c_1,\dots ,c_{2g}\mid c_1\cdots c_{2g}{c}_1^{-1}\cdots c_{2g}^{-1}⟩$ with genus $g\ge 2$, we provide an explicit bound $n-1\le \mathrm{C}\mathrm{L}(2n)=\mathrm{C}\mathrm{L}(2n+1)\le n+8g-1$ for the conjugator length function $\mathrm{C}\mathrm{L}:\mathbb{N}\to \mathbb{N}$ of ${\pi }_1({\mathrm{\Sigma }}_g)$ via a detailed analysis of conjugation reductions.