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

TL;DR

This paper develops uniform normal forms and length formulae for elements in surface groups, providing algorithms for conjugacy and root problems, with applications to growth rate computations.

Contribution

It introduces a new uniform representation of normal forms for surface group elements and derives length formulae, along with algorithms for conjugacy and root problems.

Findings

Established length inequalities for powers of elements

Derived explicit length formulae for element powers

Provided algorithms for conjugacy and root problems in surface groups

Abstract

In this paper, we primarily investigate the following symmetric presentation of the surface group $\pi_1(\Sigma_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle$. For every nontrivial element $x\in \pi_1(\Sigma_g)$, 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: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. 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 $\pi_1(\Sigma_g)$ 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.