Non-realizability of the pure braid group as area-preserving homeomorphisms

Abstract

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.