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

2025/11/17 by Ke Wang, Qiang Zhang, Wang, Ke +3
Mathematics · #Geometric and Algebraic Topology #Finite Group Theory Research #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2511.12862

Abstract

In this paper, we primarily investigate the following symmetric presentation of the surface group π₁(Σ_g)=⟨ c₁,…, c_2g| c₁⋯ c_2gc₁^-1⋯ c_2g^-1⟩. For every nontrivial element x∈ π₁(Σ_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: |x²|>|x|; |x^k|=(k-1)(|x²|-|x|)+|x|; lim_k→∞(|x^k|)/(k)=|x²|-|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 π₁(Σ_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.

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