Nielsen realization problem for the mapping class group $Mod(S_g)$ asks whether the natural projection $p_g:{Homeo}_{+}(S_g)\to Mod(S_g)$ has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion‐free subgroups of $Mod(S_g)$. The main result of this paper is that the Torelli group has no realization inside the area‐preserving homeomorphisms.

中文翻译：

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Torelli组不可实现为保留区域的同胚性

映射类组的Nielsen实现问题 $模（{\mathrm{小号}}_G）$ 询问是否自然投影 $p_G：{荷美}_{+}（{\mathrm{小号}}_G）\to 模（{\mathrm{小号}}_G）$有一节。尽管先前的所有结果都以一种必不可少的方式使用了扭转要素，但在本文中，我们将重点放在实现无扭转子组的更困难的问题上。$模（{\mathrm{小号}}_G）$。本文的主要结果是，Torelli组在保留区域的同胚性内没有实现。