Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$. All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$, the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

中文翻译：

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纯辫群作为保面积同胚的不可实现性

让$\operatorname{Homeo}_{+}(D_{n}^{2})$是 的保向同胚群$D^{2}$逐点固定边界和$n$将点标记为一组。辫群的尼尔森实现问题问是否自然投影$p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo }_{+}(D_{n}^{2}))$在子组上有一个部分$B_{n}$. 以前的所有方法都使用扭转或瑟斯顿稳定性，这不适用于纯编织组$PB_{n}$, 的子群$B_{n}$修复$n$逐点标记点。在本文中，我们证明了纯编织群在使用旋转数的保面积同胚内没有实现。