Untitled

Chen ranks and resonance varieties of the upper McCool groups

“…For each nilpotent quotient G/Γ i G, there is a filtered C-Lie algebra m(G/Γ i G), whose construction goes back to Anatoli Malcev. The Malcev Lie algebra of G is defined to be the inverse limit m(G) := lim ← − −k m(G/Γ k G), see for instance [24,26,35] for details and references. We say that the group G is 1-formal if there exists a filtered Lie algebra isomorphism between the Malcev Lie algebra m(G) and the degree completion of h(G).…”

“…The infinitesimal Alexander invariant of G is the finitely generated, graded S -module defined by B(G) := h(G) ′ /h(G) ′′ , where S = gr(R) is the symmetric algebra on H 1 (G; C). If the group G is 1-formal (in the sense of rational homotopy theory), then, as shown by Papadima and Suciu in [24], there is an isomorphism of graded S -modules, gr(B(G)) B(G). Thus, the Chen ranks of such groups G can be computed from the Hilbert series of B(G).…”

Cup products, lower central series, and holonomy Lie algebras

“…Moreover, this construction is functorial. The holonomy Lie algebra was introduced by T. Kohno in [13], building on work of Chen [6], and has been further studied in a number of papers, including [24,31,41].…”

“…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.…”

“…This work was motivated by a desire to generalize some of the results of Fenn-Sjerve [10] and Papadima-Suciu [31], from commutator-relators groups to arbitrary finitely generated groups. In [41], we studied the formality properties of finitely generated groups, focusing on the filteredformality and 1-formality properties.…”