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

Abstract

Let Homeo₊(D²_n) be the group of orientation-preserving homeomorphisms of D² 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 : Homeo₊(D²_n) → B_n : = π₀(Homeo₊(D²_n)) 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 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.

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