Word Length Formulae and Normal Forms of Conjugacy Classes in Surface Groups

In this paper, we primarily investigate the following symmetric presentation of the surface group ${\pi }_1({\mathrm{\Sigma }}_g)=⟨c_1,\dots ,c_{2g}\mid c_1\cdots c_{2g}{c}_1^{-1}\cdots c_{2g}^{-1}⟩$. For every nontrivial element $x\in {\pi }_1({\mathrm{\Sigma }}_g)$, we obtain a uniform representation of the normal forms of $x^k$ under the length-lexicographical order. Based on this, we find a new relation among these normal forms, and then derive the following three formulae related to the word length: $\mathrm{\mid }{x}^2\mathrm{\mid }>\mathrm{\mid }x\mathrm{\mid }$; $\mathrm{\mid }{x}^k\mathrm{\mid }=(k-1)(\mathrm{\mid }{x}^2\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid })+\mathrm{\mid }x\mathrm{\mid }$; ${\lim }_{k\to \mathrm{\infty }}\frac{\mathrm{\mid }{x}^k\mathrm{\mid }}{k}=\mathrm{\mid }{x}^2\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid }$. Moreover, we extend these results to obtain analogous but less precise formulae for every minimal geometric presentation. Then, we define the normal forms of conjugacy classes in ${\pi }_1({\mathrm{\Sigma }}_g)$ and give a criterion for determining the conjugacy of elements. As a consequence, we give efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications concerning the computation of some growth rates.