Braid groups are linear

By Stephen Bigelow. 2000

352 citations recorded by the source corpus.

The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a faithful representation in the case n=4. In this paper, we show that it is faithful for all n.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b3a438410234e50359dc21e2b9eb79f3531a31f8

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

DOES THE JONES POLYNOMIAL DETECT THE UNKNOT (2000, background): “om d0 to another point on ∂D n, • Nmeets ∂D n only at its endpoints, and • a component of D n \Ncontains precisely one puncture point. This last requirement was not included in the deﬁnition given in [3]. Without it, Theorem 5.1 is not true, as far as I know. Let Fbe a fork and let Nbe a noodle. We ”
Braid groups and Hodge theory (2013, background): “X)q. It would be interesting to study the associated period maps as Hodge-theoretic counterparts to the Lawrence– Krammer–Bigelow representations ξq of Bn [Law], [At], [Kr], [ Bg2 ].”
Linearity of artin groups of finite type (2000, background): “Recently, both Bigelow [1] and Krammer [6] proved that the braid groups are linear.”
Artin monoids inject in their groups (2001, methodology): “d we prove in Section 4 that ψis injective. If Γ = A n, then G A n is the braid group on n+ 1 strings, and ψ: G + A n → Gl(V) is equivalent to the representation constructed by Bigelow and Krammer in [Big], [Kra1] and [Kra2]. In this case, V has ﬁnite dimension, and the injectivity of ψimplies the i”
Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group (2008, background): “. It is not known whether mapping class groups are linear, except in special cases: most notably, Krammer and Bigelow proved linearity of braid groups (mapping class groups of punctured spheres) [K], [Bi]. However, many linear group results inspire mapping class group analogues. For example, McCarth”
Complete positive group presentations (2001, background): “6? Similarly, but on the other direction, it is known that every Artin monoid embeds in the corresponding group [29], but the remarkable proof of the result uses an indirect approach via a linear representation (inspired by [25]).”
An introduction to right-angled Artin groups (2006, background): “rical Artin groups are known to have the following properties. (1) There exist ﬁnite K(A,1)-spaces. [28], [17] (2) A is torsion-free. [21] (3) The center of A is inﬁnite cyclic. [21] (4) A is linear. [11], [63], [32] (5) A is biautomatic and hence has solvable word and conjugacy problem. [43], [21],”
Advances in Cryptology - CRYPTO 2003 (2003, methodology): “[13,11,12] given the first formal model of security and the first provablysecure protocols for the group AKE setting.”
Problems on invariants of knots and 3-manifolds (2004, background): “Krammer’s proof of this fact [229] works when t is assigned any value between 0 and 1.”
Automorphism groups of free groups, surface groups and free abelian groups (2005, background): “Note that braid groups are linear [8] but it is unknown if mapping class groups of closed surfaces are.”
Foundations of Garside Theory (2013, methodology): “…normal form , or Thurston’s normal form —it became a standard tool in the investigation of braids and Artin–Tits monoids and groups from a viewpoint of geometric group theory and representation, essential in particular in Krammer’s algebraic proof of the linearity of braid groups [149, 150].”