Resonance of basis-conjugating automorphism groups

Abstract. We investigate the relationship between the geometric Bieri-NeumannStrebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as Kähler and quasi-Kähler manifolds.

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“…For each a ∈ A 1 , we then have a 2 = 0. Thus, right-multiplication by a defines a cochain complex (14) (A, ·a) :…”

Section: Resonance Varieties Formality and The Exponential Formulamentioning

confidence: 99%

Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

Papadima¹,

Suciu

2009

Proceedings of the London Mathematical Society

144

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“…The resonance varieties of a variety spaces and groups have been studied intensively from many perspectives and in varying degrees of generality, see for instance [32,68,60] and reference therein. In particular, the first resonance varieties of the groups P n , wP n , and wP + n have been completely determined in [27,21,71]. We summarize those results as follows.…”

Section: Resonance Varietiesmentioning

confidence: 85%

“…The welded pure braid group wP n , also known as the group of basis-conjugating automorphisms in [4,37,21], or the McCool group in [15], is defined as…”

Section: Welded Braid Groupsmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

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The Pure Braid Groups and Their Relatives

Suciu

Wang

2017

Perspectives in Lie Theory

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In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups P n , the welded pure braid groups wP n , the virtual pure braid groups vP n , as well as their 'upper' variants, wP + n and vP + n . We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.

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Chen ranks and resonance varieties of the upper McCool groups

Suciu

Wang

2019

Advances in Applied Mathematics

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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.

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Resonance of basis-conjugating automorphism groups

Abstract: Abstract. We determine the structure of the first resonance variety of the cohomology ring of the group of automorphisms of a finitely generated free group which act by conjugation on a given basis.

Cited by 4 publications

References 13 publications

Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

The Pure Braid Groups and Their Relatives

Chen ranks and resonance varieties of the upper McCool groups

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