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

Part of: Smooth dynamical systems: general theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  11 June 2020

LEI CHEN Open the ORCID record for LEI CHEN [Opens in a new window]
LEI CHEN*
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA91125, USA email [email protected]
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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.

Keywords

group actionslow-dimensional dynamicstopological dynamics

MSC classification

Secondary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37C25: Fixed points, periodic points, fixed-point index theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 1988 - 1999
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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