Word Length Formulae and Normal Forms of Conjugacy Classes in Surface Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909976593694720 |
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| author | Wang, Ke Zhang, Qiang Zhao, Xuezhi |
| author_facet | Wang, Ke Zhang, Qiang Zhao, Xuezhi |
| contents | In this paper, we primarily investigate the following symmetric presentation of the surface group $π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle$. For every nontrivial element $x\in π_1(Σ_g)$, 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: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. 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 $π_1(Σ_g)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_12862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Word Length Formulae and Normal Forms of Conjugacy Classes in Surface Groups Wang, Ke Zhang, Qiang Zhao, Xuezhi Geometric Topology Group Theory 20E45, 20F06, 20F10, 13P10 In this paper, we primarily investigate the following symmetric presentation of the surface group $π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle$. For every nontrivial element $x\in π_1(Σ_g)$, 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: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. 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 $π_1(Σ_g)$ 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. |
| title | Word Length Formulae and Normal Forms of Conjugacy Classes in Surface Groups |
| topic | Geometric Topology Group Theory 20E45, 20F06, 20F10, 13P10 |
| url | https://arxiv.org/abs/2511.12862 |