AI 中文总结
研究闭可定向曲面上简单闭曲线，通过元素在到有限群满同态下的像特征，证明简单闭曲线集合在有限拓扑中是闭集，得到判定元素能否由简单闭曲线表示的新算法，还讨论了相关特殊情形。
AI 中文摘要
本文建立了闭可定向曲面上简单闭曲线的新特征。设$\Gamma$为闭可定向曲面的基本群。我们证明，若元素$g\in\Gamma$在从$\Gamma$到每个有限群的满同态下与给定简单闭曲线$\gamma\in\Gamma$有相同的可能像，则$g$属于$\gamma$的$\mathrm{Aut}(\Gamma)$轨道，即$g$本身是与$\gamma$具有相同拓扑类型的简单闭曲线。因此，$\Gamma$中简单闭曲线的集合在$\Gamma$的有限拓扑中是闭集；我们得到了一种新算法来判定$\Gamma$中给定元素是否可由简单闭曲线表示。还讨论了简单闭曲线的真幂次和pro - $p$情形。
英文摘要
This paper establishes a new characterization of simple closed curves on a closed orientable surface. Let $Γ$ be the fundamental group of a closed orientable surface. We prove that if an element $g\inΓ$ has the same possible images as a given simple closed curve $γ\in Γ$ under epimorphisms from $Γ$ to every finite group, then $g$ belongs to the $\mathrm{Aut}(Γ)$-orbit of $γ$, i.e. $g$ is itself a simple closed curve with the same topological type as $γ$. Consequently, the set of simple closed curves in $Γ$ is closed in the profinite topology of $Γ$; and we obtain a new algorithm to decide whether a given element in $Γ$ can be represented by a simple closed curve. Proper powers of simple closed curves and the pro-$p$ cases are also discussed.
Comments24 pages, 2 figures
