On injective homomorphisms for pure braid groups, and associated Lie algebras

Abstract

The purpose of this article is to record the center of the Lie algebra obtained from the descending central series of Artin's pure braid group, a Lie algebra analyzed in work of Kohno [T. Kohno, Linear representations of braid groups and classical Yang-T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985) 57-75], and Falk and Randell [M. Falk, R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985) 77-88]. The structure of this center gives a Lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criterion used to test whether certain morphisms out of free groups are faithful [F.R. Cohen, J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, in: Algebraic Topology: Categorical Decomposition Techniques, in: Progr. Math., vol. 215, Birkhäuser, Basel, 2003; Braid groups, free groups, and the loop space of the 2-sphere, math.AT/0409307]. However, it is as unclear whether this criterion for faithfulness can be applied to any open cases concerning representations of P n such as the Gassner representation.

On braid groups and homotopy groups

“…Certain Galois groups G are identified as natural subgroups of these automorphism groups by Belyȋ [2], Deligne [12], Drinfel'd [13,14], Ihara [20,21] and Schneps [33]. for which the kernel of Ad is precisely the center of gr * (P n+1 ) (Cohen and Prassidis [8]). Combining this last fact with Theorem 3.2 gives properties of the composite morphism of Lie algebras…”

Remarks Concerning Lubotzky's Filtration

“…It is shown below that G is sometimes linear. Thus it is natural to ask the following question which is also raised in [2] with some additional evidence here. with F n a free group on n letters and Γ a group that admits a finite dimensional faithful linear representation.…”

“…(1) The subset F j (G) is naturally a subgroup of G. (2) There is a morphism of split group extensions…”

On the kernel of the Gassner representation

“…P r o o f. A proof of the first equivalence may be found in, for example, [3]. The second equivalence is an easy exercise about free groups and is left to the reader.…”

“…, t ±1 n ]). The faithfulness of G n would follow from that of g n (see Proposition 2.1 below, or [3] for a more general result).…”