A RRANGEMENT GROUPS , LOWER CENTRAL SERIES , AND M ASSEY PRODUCTS Alex Suciu Northeastern University Workshop on Hyperplane Arrangements Institute of Mathematics Vietnam Academy of Science and Technology March 22, 2019 A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 1 / 22
L IE ALGEBRAS ATTACHED TO GROUPS L OWER CENTRAL SERIES L OWER CENTRAL SERIES Let G be a group. The lower central series t γ k p G qu k ě 1 is defined inductively by γ 1 p G q “ G and γ k ` 1 p G q “ r G , γ k p G qs . Here, if H , K ă G , then r H , K s is the subgroup of G generated by tr a , b s : “ aba ´ 1 b ´ 1 | a P H , b P K u . If H , K Ÿ G , then r H , K s Ÿ G . The subgroups γ k p G q are, in fact, characteristic subgroups of G . Moreover r γ k p G q , γ ℓ p G qs Ď γ k ` ℓ p G q , @ k , ℓ ě 1. γ 2 p G q “ r G , G s is the derived subgroup, and so G { γ 2 p G q “ G ab . r γ k p G q , γ k p G qs Ÿ γ k ` 1 p G q , and thus the LCS quotients, gr k p G q : “ γ k p G q{ γ k ` 1 p G q are abelian. If G is finitely generated, then so are its LCS quotients. Set φ k p G q : “ rank gr k p G q . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 2 / 22
L IE ALGEBRAS ATTACHED TO GROUPS A SSOCIATED GRADED L IE ALGEBRA A SSOCIATED GRADED L IE ALGEBRA Fix a coefficient ring k . Given a group G , we let à gr p G , k q “ gr k p G q b k . k ě 1 This is a graded Lie algebra, with Lie bracket r , s : gr k ˆ gr ℓ Ñ gr k ` ℓ induced by the group commutator. For k “ Z , we simply write gr p G q “ gr p G , Z q . The construction is functorial. Example: if F n is the free group of rank n , then gr p F n q is the free Lie algebra Lie p Z n q . ř k gr k p F n q is free abelian, of rank φ k p F n q “ 1 d . d | k µ p d q n k A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 3 / 22
L IE ALGEBRAS ATTACHED TO GROUPS C HEN L IE ALGEBRAS C HEN L IE ALGEBRAS Let G p i q be the derived series of G , starting at G p 1 q “ G 1 , G p 2 q “ G 2 , and defined inductively by G p i ` 1 q “ r G p i q , G p i q s . The quotient groups, G { G p i q , are solvable; G { G 1 “ G ab , while G { G 2 is the maximal metabelian quotient of G . The i-th Chen Lie algebra of G is defined as gr p G { G p i q , k q . Clearly, this construction is functorial. The projection q i : G ։ G { G p i q , induces a surjection gr k p G ; k q ։ gr k p G { G p i q ; k q , which is an iso for k ď 2 i ´ 1. Assuming G is finitely generated, write θ k p G q “ rank gr k p G { G 2 q for the Chen ranks . We have φ k p G q ě θ k p G q , with equality for k ď 3. ` n ` k ´ 2 ˘ Example (K.-T. Chen 1951): θ k p F n q “ p k ´ 1 q , for k ě 2. k A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 4 / 22
L IE ALGEBRAS ATTACHED TO GROUPS H OLONOMY L IE ALGEBRA H OLONOMY L IE ALGEBRA A quadratic approximation of the Lie algebra gr p G , k q , where k is a field, is the holonomy Lie algebra of G , which is defined as h p G , k q : “ Lie p H 1 p G , k qq{x im p µ _ G qy , where L “ Lie p V q the free Lie algebra on the k -vector space V “ H 1 p G ; k q , with L 1 “ V and L 2 “ V ^ V . µ _ G : H 2 p G , k q Ñ V ^ V is the dual of the cup product map µ G : H 1 p G ; k q ^ H 1 p G ; k q Ñ H 2 p G ; k q . There is a surjective morphism of graded Lie algebras, � � gr p G ; k q , h p G , k q (*) which restricts to isomorphisms h k p G , k q Ñ gr k p G ; k q for k ď 2. A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 5 / 22
L IE ALGEBRAS ATTACHED TO GROUPS A RRANGEMENT GROUPS AND L IE ALGEBRAS A RRANGEMENT GROUPS AND L IE ALGEBRAS Let A “ t ℓ 1 , . . . , ℓ n u be an affine line arrangement in C 2 , and let G “ G p A q be the fundamental group of the complement of A . The holonomy Lie algebra h p A q : “ h p G p A qq has (combinatorially determined) presentation ÿ @ D r x j , x k s , j P p h p A q “ x 1 , . . . , x n | P , P P P k P P where x i represents the meridian about the i -th line, P Ă 2 r n s is the set of multiple points, and p P “ P zt max P u for P P P . Thus, every double point P “ L i X L j contributes a relation r x i , x j s , each triple point P “ L i X L j X L k contributes two relations, r x i , x j s ` r x i , x k s and ´r x i , x j s ` r x j , x k s , etc. Consequently, h 1 p A q is free abelian with basis t x 1 , . . . , x n u , while ` n ˘ ´ ř h 2 p A q is free abelian of rank φ 2 “ P P P p| P | ´ 1 q , with basis 2 tr x i , x j s : i , j P p P , P P P u . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 6 / 22
L IE ALGEBRAS ATTACHED TO GROUPS A RRANGEMENT GROUPS AND L IE ALGEBRAS The canonical projection h p G , Q q ։ gr p G , Q q is an isomorphism. Thus, the LCS ranks φ k p G q are combinatorially determined. (Falk–Randell 1985) If A is supersolvable , with exponents d 1 , . . . , d ℓ , then G “ F d ℓ ¸ ¨ ¨ ¨ ¸ F d 2 ¸ F d 1 (almost direct product) and ÿ ℓ φ k p G q “ φ k p F d i q . i “ 1 (Papadima–S. 2006) If A is decomposable , then h p G q ։ gr p G q is an isomorphism, and gr k p G q is free abelian of rank ÿ φ k p G q “ φ k p F µ p X q q for k ě 2 . X P L 2 p A q (S. 2001) For G “ G p A q , the groups gr k p G q may have non-zero torsion. Question: Is that torsion combinatorially determined? (Artal Bartolo, Guerville-Ballé, and Viu-Sos 2018): Answer: No! A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 7 / 22
F ORMALITY PROPERTIES M ALCEV L IE ALGEBRA M ALCEV L IE ALGEBRA Let k be a field of characteristic 0. The group-algebra k G has a natural Hopf algebra structure, with comultiplication ∆ p g q “ g b g and counit ε : k G Ñ k . Let I “ ker ε . The I -adic completion x Ý k k G { I k is a filtered, k G “ lim Ð complete Hopf algebra. An element x P x k G is called primitive if p ∆ x “ x p b 1 ` 1 p b x . The set of all such elements, m p G , k q “ Prim p x k G q , with bracket r x , y s “ xy ´ yx , is a complete, filtered Lie algebra, called the Malcev Lie algebra of G . If G is finitely generated, then m p G , k q “ lim Ý k L p G { γ k p G q b k q , and Ð gr p m p G , k qq – gr p G , k q . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 8 / 22
F ORMALITY PROPERTIES F ORMALITY AND FILTERED FORMALITY F ORMALITY AND FILTERED FORMALITY Let G be a finitely generated group, k a field of characteristic 0. G is filtered-formal (over k ), if there is an isomorphism of filtered Lie algebras, m p G ; k q – p gr p G ; k q . G is 1 -formal (over k ) if it is filtered formal and the canonical projection h p G , k q ։ gr p G ; k q is an isomorphism; that is, m p G ; k q – p h p G ; k q . An obstruction to 1-formality is provided by the Massey products x α 1 , α 2 , α 3 y P H 2 p G , k q , for α i P H 1 p G , k q with α 1 α 2 “ α 2 α 3 “ 0. T HEOREM (S.–W ANG ) The above formality properties are preserved under finite direct products and coproducts, split injections, passing to solvable quotients, as well as extension or restriction of coefficient fields. A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 9 / 22
F ORMALITY PROPERTIES F ORMALITY AND FILTERED FORMALITY Examples of 1-formal groups Fundamental groups of compact Kähler manifolds; e.g., surface groups. Fundamental groups of complements of complex algebraic affine hypersurfaces; e.g., arrangement groups, free groups. Right-angled Artin groups. Examples of filtered formal groups Finitely generated, torsion-free, 2-step nilpotent groups with torsion-free abelianization; e.g., the Heisenberg group. Fundamental groups of Sasakian manifolds. Fundamental groups of graphic configuration spaces of surfaces of genus g ě 1; e.g., pure braid groups of elliptic curves. Examples of non-filtered formal groups Certain finitely generated, torsion-free, 3-step nilpotent groups. A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 10 / 22
F ORMALITY PROPERTIES C HEN L IE ALGEBRAS AND FILTERED FORMALITY C HEN L IE ALGEBRAS AND FILTERED FORMALITY T HEOREM (P APADIMA –S., S.–W ANG ) For each i ě 2 , there is an isomorphism of complete, separated, filtered Lie algebras, m p G { G p i q ; k q – m p G ; k q{ m p G ; k q p i q . T HEOREM (SW) For each i ě 2 , the quotient map G ։ G { G p i q induces a natural epimorphism of graded k -Lie algebras, � � gr p G { G p i q ; k q . gr p G ; k q{ gr p G ; k q p i q Moreover, if G is filtered formal, this map is an isomorphism and G { G p i q is also filtered formal. A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT GROUPS , LCS & M ASSEY M ARCH 22, 2019 11 / 22
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