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

Abstract: For a surface group $π1(Σ_g)=\langle c_1,\dots , c{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\rangle$ with genus $g\geq 2$, we provide an explicit bound $n-1\leq \mathrm{CL}(2n)=\mathrm{CL}(2n+1)\leq n+8g-1$ for the conjugator length function $\mathrm{CL}:\mathbb N\to\mathbb N$ of $π_1(Σ_g)$ via a detailed analysis of conjugation reductions.