Free groups and finite-type invariants of pure braids (Q2777978)

The well-known theory of Vassiliev invariants of knots can be extended to pure braids. As with the case of knots, a singular pure braid is a pure braid, with the exception that its strands are allowed to intersect transversally in a finite number of ``double points'' (so a pure braid is just a singular pure braid with no double points). Let \(P_k\) denote the pure braid group on \(k\) strands and let \(R\) be a commutative ring with a 1. A pure braid invariant \(v \colon P_k \to R\) can be extended inductively to an invariant of singular pure braids by means of the Vassiliev skein relation. An invariant of pure braids is said to be of type \(n\) if its extension vanishes on all singular pure braids with more than \(n\) double points. An invariant is said to be a finite-type or Vassiliev invariant if it is of type \(n\) for some \(n\). Also by using the Vassiliev skein relation, a singular braid with \(k\) strands can be formally considered as an element of the group algebra \(RP_k\), and then the extension of an invariant to singular braids is precisely the same as the linear extension of the invariant to the group algebra. NEWLINENEWLINENEWLINELet \(J^nP_k\) denote the \(n\)th power of the augmentation ideal of \(RP_k\). Then it is shown that \(J^nP_k\) is spanned by singular braids with \(n\) double points. It follows that the type \(n\) invariants of \(P_k\) can be identified with the \(R\)-module \(\text{Hom}_R (RP_k/J^{n+1}P_k,R)\). Therefore the study of the type \(n\) invariants of \(P_k\) has been reduced to a purely algebraic problem. NEWLINENEWLINENEWLINELet \(F_i\) denote the free group on \(i\) generators and let \(\prod_k\) denote the direct product \(F_{k-1} \times \dots \times F_2 \times F_1\). Then \(P_k\) is the semidirect product \(F_{k-1} \ltimes P_{k-1}\), so by induction is an iterated semidirect product \(F_{k-1} \ltimes \dots \ltimes F_2 \ltimes F_1\). Therefore there is a bijection (but not a homomorphism) \(E \colon P_k \to \prod_k\) and this extends to an \(R\)-linear map \(E \colon RP_k \to R\prod_k\). The theory of Artin combing of braids is now used to deduce that \(E\) induces an \(R\)-module isomorphism \(J^n E_k \to J^n \prod_k\) for all \(n \geq 0\). NEWLINENEWLINENEWLINEFor any group \(G\), set \(A(G) = \widehat{\bigoplus}_{n\geq 0} J^nG/J^{n+1}G\), the completion of the direct sum with respect to the grading. In the case of free groups, it is well known that \(A(F_i) \cong R[[X_1,\dots,X_i]]\). Using the \(R\)-module isomorphism of the previous paragraph, we obtain an \(R\)-isomorphism NEWLINE\[NEWLINE A(P_k) \longrightarrow A(F_{k-1}) \otimes_R \dots \otimes_R A(F_2) \otimes_R A(F_1). NEWLINE\]NEWLINE This has various consequences; for example in the case \(R= {\mathbb Z}\), we see that \(J^nG/J^{n+1}G\) is torsion free for all \(n\). Also the Vassiliev invariants can be considered as elements of \(\text{Hom}_R(RP_k,R)\). NEWLINENEWLINENEWLINEIn this paper, properties of the Magnus expansion are used to derive the existence of a universal Vassiliev invariant \(U : RP_k \to A(P_k)\). This is an \(R\)-map which has the property that for any \(R\)-valued finite type invariant \(v\) (considered as an element of \(\text{Hom}_R(RP_k,R)\)), there is a unique \(R\)-map \(\widehat{v} : A(P_k) \to R\) such that \(\widehat{v} \circ U = v\). NEWLINENEWLINENEWLINEAn \(n\)-trivial braid is one which cannot be distinguished from the trivial braid by invariants of type less than \(n\). Then combining the above methods with properties of the dimension subgroup, another result obtained is that the \(n\)-trivial braids are precisely those in the \(n\)th subgroup \(\gamma_nP_k\) of the lower central series. This result has already been obtained by \textit{Toshitake Kohno} [Contemp. Math. 179, 123-138 (1994; Zbl 0876.57009)]. NEWLINENEWLINENEWLINEIn the penultimate section of the paper, calculations are made on the number of linearly independent invariants of each type for each pure braid group. This amounts to finding the dimensions of each \(J^nP_k/J^{n+1}P_k\); such computations were done by \textit{A. Stoimenow} [On Harrison cohomology and a conjecture by Drinfel'd. Monograph Freie Universität, Berlin 1996]. NEWLINENEWLINENEWLINEAs the authors state, many of the results of this paper are known. However the main technical tool, the use of Artin combing to identify powers of the augmentation ideal of pure braid groups with those of products of free groups, appears to be new.