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The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

The Algebraic Braid Groups are Torsion-Free: an Algebraic Proof.

Joan L. Dyer (1980)

Mathematische Zeitschrift

The braid groups of the projective plane.

Gonçalves, Daciberg Lima, Guaschi, John (2004)

Algebraic & Geometric Topology

The Burau representation is not faithful for $n = 5$ .

Bigelow, Stephen (1999)

Geometry & Topology

The dual braid monoid

David Bessis (2003)

Annales scientifiques de l'École Normale Supérieure

The fundamental group of a Galois cover of $ℂ ℙ^{1} \times T$ .

Amram, Meirav, Goldberg, David, Teicher, Mina, Vishne, Uzi (2002)

Algebraic & Geometric Topology

The Homological Algebra of Artin Groups.

Craig C. Squier (1994)

Mathematica Scandinavica

The Hurwitz action and braid group orderings.

Funk, Jonathon (2001)

Theory and Applications of Categories [electronic only]

The interplay between linear representations of the braid group.

Abdulrahim, Mohammad N., Kassem, Nibal H. (2007)

International Journal of Mathematics and Mathematical Sciences

The lower central series of a fiber-type arrangement.

M., Randell, R. Falk (1985)

Inventiones mathematicae

The Magic World of Geometry - II. Geometry and Algebra of Braids.

Vagn Lundsgaard Hansen (1994)

Elemente der Mathematik

The Magic World of Geometry - III. The Dirac String Problem.

Vagn Lundsgaard Hansen (1994)

Elemente der Mathematik

The mapping class group of a genus two surface is linear.

Bigelow, Stephen J., Budney, Ryan D. (2001)

Algebraic & Geometric Topology

The $n$ th root of a braid is unique up to conjugacy.

Gonzalez-Meneses, Juan (2003)

Algebraic & Geometric Topology

The orbits of the Hurwitz action of the braid groups on the standard generators

Yoshiro Yaguchi (2010)

Fundamenta Mathematicae

The Hurwitz action of the n-braid group Bₙ on the n-fold direct product $(B_{m}) ⁿ$ of the m-braid group $B_{m}$ is studied. We show that the orbit of any n- tuple of the n standard generators of $B_{n + 1}$ consists of the (n-1)th powers of n+1 elements.

The proof of Birman's conjecture on singular braid monoids.

Paris, Luis (2004)

Geometry & Topology

The virtual and universal braids

Valerij G. Bardakov (2004)

Fundamenta Mathematicae

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi-direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi-direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as quotients the singular braid group, virtual...

The Yang-Baxter equation and invariants of links.

V.G. Turaev (1988)

Inventiones mathematicae

The Yokonuma-Temperley-Lieb algebra

D. Goundaroulis, J. Juyumaya, A. Kontogeorgis, S. Lambropoulou (2014)

Banach Center Publications

We define the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra, leading to a sequence of knot invariants which coincide with the Jones polynomial.

Topics in statistical physics involving braids

J. McCabe, T. Wydro (1998)

Banach Center Publications

We review the appearance of the braid group in statistical physics. In particular, we explain its relevance to the anyon model of fractional statistics and conformal field theory.

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