Chen ranks and resonance varieties of the upper McCool groups

Abstract

The group of basis-conjugating automorphisms of the free group of rank n, also known as the McCool group or the welded braid group PΣ n , contains a much-studied subgroup, called the upper McCool group PΣ + n . Starting from the cohomology ring of PΣ + n , we find, by means of a Gröbner basis computation, a simple presentation for the infinitesimal Alexander invariant of this group, from which we determine the resonance varieties and the Chen ranks of the upper McCool groups. These computations reveal that, unlike for the pure braid group P n and the full McCool group PΣ n , the Chen ranks conjecture does not hold for PΣ + n , for any n ≥ 4. Consequently, PΣ + n is not isomorphic to P n in that range, thus answering a question of Cohen, Pakianathan, Vershinin, and Wu. We also determine the scheme structure of the resonance varieties R 1 (PΣ + n ), and show that these schemes are not reduced for n ≥ 4.

The Pure Braid Groups and Their Relatives

“…Furthermore, as shown in [11,50], the groups vP n and vP + n are graded-formal. On the other hand, we show in [71] that the virtual pure braid groups vP n and vP + n are 1-formal if and only if n ≤ 3. A lot is also known about the residual properties of the pure braid-like groups, especially as they relate to the lower central series.…”

“…Resonance-Chen ranks formula P n n 3 + n 4 planes [27] (k − 1) n+1 4 [24] Yes [27] wP n n 2 planes and n 4 linear spaces of dimension 3 [21] (k − 1) n 2 + (k 2 − 1) n 3 for k 3 [23] Yes [23] wP + n (n − i) linear spaces of dimension i ≥ 2 [71] k ∑ i=3 n+i−2 i+1 + n+1 4 [71] No [71] vP 3 H 1 (vP 3 , C) = C 6 [10,70] k+3 [70] No [70] vP + 4 3-dimensional non-linear subvariety of degree 6 [70] (k 3 − 1) + k 2 [70] No [70] basis, and so the algebra A is Koszul. For more details and references regarding this topic, we direct the reader to Table 1 and to §3.1.…”

“…The aforementioned computations of the cohomology algebras of the various pure braid-like groups allows one to determine the corresponding resonance varieties, at least in principle. In the case of the first resonance varieties of the groups P n , wP n , and wP + n , complete answers can be found in [27], [21], and [71], respectively, while for the virtual pure braid groups, partial answers are given in [70]. We list some of the features of these varieties in Table 2.…”

Cup products, lower central series, and holonomy Lie algebras

“…In the special case when G admits a presentation with only commutator relators, presentations for these Lie algebras were given by Papadima and Suciu in [31]. For arbitrary finitely generated groups G, the metabelian quotient h(G)/h(G) ′′ , also known as the holonomy Chen Lie algebra of G, is closely related to the first resonance variety of G, a geometric object which has been studied intensely from many points of view, see for instance [27,34,35,43,44] and references therein.…”

“…In related work, we apply the techniques developed in this paper and in [41] to the study of several families of "pure-braid like" groups. For instance, we investigate in [43] the pure virtual braid groups, and we investigate in [44] the McCool groups, also known as the pure welded braid groups. A summary of these results, as well as further motivation and background can be found in [42].…”