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

Abstract

Let ${\text{Homeo}}_{+}(D_n^2)$ be the group of orientation-preserving homeomorphisms of $D^2$ fixing the boundary pointwise and $n$ marked points as a set. Nielsen realization problem for the braid group asks whether the natural projection $p_n:{\text{Homeo}}_{+}(D_n^2)\to B_n:={\pi }_0({\text{Homeo}}_{+}(D_n^2))$ has a section over subgroups of $B_n$. All of the previous methods either use torsions or Thurston stability, which do not apply to the pure braid group $P{B}_n$, 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.

12 pages. arXiv admin note: text overlap with arXiv:1904.09490