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

2020 ◽  
pp. 1-12
Author(s):  
LEI CHEN
Keyword(s):  
Braid Group ◽  
Natural Projection ◽  
Realization Problem ◽  
Pure Braid Group ◽  
Rotation Numbers ◽  
Marked Points ◽  
Area Preserving ◽  
Nielsen Realization

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.

scholarly journals Representation Stability of Power Sets and Square Free Polynomials

2015 ◽  
Vol 67 (5) ◽  
pp. 1024-1045
Author(s):  
Samia Ashraf ◽  
Haniya Azam ◽  
Barbu Berceanu
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Point Of View ◽  
Pure Braid Group ◽  
Representation Stability ◽  
Stability Point ◽  
Power Set ◽  
The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

scholarly journals Classifying links under fused isotopy

2016 ◽  
Vol 25 (07) ◽  
pp. 1650042
Author(s):  
Andrew Fish ◽  
Ebru Keyman
Keyword(s):  
Braid Group ◽  
Algebraic Proof ◽  
Pure Braid Group ◽  
Linking Numbers

All knots have been shown to be isotopic to the unknot using a process known as virtualization. We extend and adapt this process to show that, up to fused isotopy, classical links are classified by their linking numbers. We provide an algebraic proof, utilizing Alexander’s Theorem and some simple results about the pure braid group.

scholarly journals Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces

2015 ◽  
Vol 26 (09) ◽  
pp. 1550066 ◽  
Author(s):  
Michael Brandenbursky
Keyword(s):  
Braid Group ◽  
Infinite Family ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Invariant Metrics ◽  
Closed Orientable Surface ◽  
Infinite Dimensional ◽  
Hamiltonian Diffeomorphisms ◽  
Injective Homomorphisms ◽  
Area Preserving

Let Σg be a closed orientable surface of genus g and let Diff 0(Σg, area ) be the identity component of the group of area-preserving diffeomorphisms of Σg. In this paper, we present the extension of Gambaudo–Ghys construction to the case of a closed hyperbolic surface Σg, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Σg defines a nontrivial homogeneous quasi-morphism on the group Diff 0(Σg, area ). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff 0(Σg, area ) is infinite-dimensional. Let Ham (Σg) be the group of Hamiltonian diffeomorphisms of Σg. As an application of the above construction we construct two injective homomorphisms Zm → Ham (Σg), which are bi-Lipschitz with respect to the word metric on Zm and the autonomous and fragmentation metrics on Ham (Σg). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham (Σg).

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