F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY AND THREE - DIMENSIONAL TOPOLOGY Alex Suciu Northeastern University Colloquium University of Fribourg June 7, 2016 A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 1 / 26
C AST OF CHARACTERS F UNDAMENTAL GROUPS OF MANIFOLDS F UNDAMENTAL GROUPS OF MANIFOLDS Every finitely presented group π can be realized as π = π 1 ( M ) , for some smooth, compact, connected manifold M n of dim n ě 4. M n can be chosen to be orientable. If n even, n ě 4, then M n can be chosen to be symplectic (Gompf). If n even, n ě 6, then M n can be chosen to be complex (Taubes). Requiring that n = 3 puts severe restrictions on the (closed) 3-manifold group π = π 1 ( M 3 ) . A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 2 / 26
C AST OF CHARACTERS K ÄHLER GROUPS K ÄHLER GROUPS A Kähler manifold is a compact, connected, complex manifold, with a Hermitian metric h such that ω = im ( h ) is a closed 2-form. Smooth, complex projective varieties are Kähler manifolds. A group π is called a Kähler group if π = π 1 ( M ) , for some Kähler manifold M . The group π is a projective group if M can be chosen to be a projective manifold. The classes of Kähler and projective groups are closed under finite direct products and passing to finite-index subgroups. Every finite group is a projective group. [Serre „ 1955] A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 3 / 26
C AST OF CHARACTERS K ÄHLER GROUPS The Kähler condition puts strong restrictions on π , e.g.: π is finitely presented. b 1 ( π ) is even. [by Hodge theory] π is 1-formal [Deligne–Griffiths–Morgan–Sullivan 1975] (i.e., its Malcev Lie algebra m ( π ) : = Prim ( z Q [ π ]) is quadratic) π cannot split non-trivially as a free product. [Gromov 1989] Problem: Are all Kähler groups projective groups? Problem [Serre]: Characterize the class of projective groups. A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 4 / 26
C AST OF CHARACTERS Q UASI - PROJECTIVE GROUPS Q UASI - PROJECTIVE GROUPS A group π is said to be a quasi-projective group if π = π 1 ( M z D ) , where M is a smooth, projective variety and D is a divisor. Qp groups are finitely presented. The class of qp groups is closed under direct products and passing to finite-index subgroups. For a qp group π , b 1 ( π ) can be arbitrary (e.g., the free groups F n ). π may be non-1-formal (e.g., the Heisenberg group). π can split as a non-trivial free product (e.g., F 2 = Z ˚ Z ). A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 5 / 26
C AST OF CHARACTERS C OMPLEMENTS OF HYPERSURFACES C OMPLEMENTS OF HYPERSURFACES A subclass of quasi-projective groups consists of fundamental groups of complements of hypersurfaces in CP n , π = π 1 ( CP n zt f = 0 u ) , f P C [ z 0 , . . . , z n ] homogeneous . All such groups are 1-formal. [Kohno 1983] By the Lefschetz hyperplane sections theorem, π = π 1 ( CP 2 z C ) , for some plane algebraic curve C . Zariski asked Van Kampen to find presentations for such groups. Using the Alexander polynomial, Zariski showed that π is not determined by the combinatorics of C , but depends on the position of its singularities. P ROBLEM (Z ARISKI ) Is π = π 1 ( CP 2 z C ) residually finite , i.e., is the map to the profinite completion, π Ñ π alg : = lim Ý G Ÿ f . i . π π / G, injective? Ð A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 6 / 26
C AST OF CHARACTERS L INE ARRANGEMENTS H YPERPLANE ARRANGEMENTS Even more special are the arrangement groups , i.e., the fundamental groups of complements of complex hyperplane arrangements (or, equivalently, complex line arrangements). Let A be an arrangement of lines in CP 2 , defined by a polynomial f = ś L P A f L , with f L linear forms so that L = P ( ker ( f L )) . The combinatorics of A is encoded in the intersection poset , L ( A ) , with L 1 ( A ) = t lines u and L 2 ( A ) = t intersection points u . P 1 P 2 P 3 P 4 L 4 L 3 P 4 L 2 P 3 L 1 P 1 P 2 L 1 L 2 L 3 L 4 A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 7 / 26
C AST OF CHARACTERS L INE ARRANGEMENTS Let U ( A ) = CP 2 z Ť L P A L . The group π = π 1 ( U ( A )) has a finite presentation with Meridional generators x 1 , . . . , x n , where n = | A | , and ś x i = 1. Commutator relators x i α j ( x i ) ´ 1 , where α 1 , . . . α s P P n Ă Aut ( F n ) , and s = | L 2 ( A ) | . Let γ 1 ( π ) = π , γ 2 ( π ) = π 1 = [ π , π ] , γ k ( π ) = [ γ k ´ 1 ( π ) , π ] , be the lower central series of π . Then: π ab = π / γ 2 equals Z n ´ 1 . π / γ 3 is determined by L ( A ) . π / γ 4 (and thus, π ) is not determined by L ( A ) (G. Rybnikov). P ROBLEM (O RLIK ) Is π torsion-free? Answer is yes if U ( A ) is a K ( π , 1 ) . This happens if the cone on A is a simplicial arrangement (Deligne), or supersolvable (Terao). A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 8 / 26
C AST OF CHARACTERS A RTIN GROUPS A RTIN GROUPS Let Γ = ( V , E ) be a finite, simple graph, and let ℓ : E Ñ Z ě 2 be an edge-labeling. The associated Artin group : A Γ , ℓ = x v P V | vwv ¨ ¨ ¨ = wvw ¨ ¨ ¨ , for e = t v , w u P E y . loomoon looomooon ℓ ( e ) ℓ ( e ) If ( Γ , ℓ ) is Dynkin diagram of type A n ´ 1 with ℓ ( t i , i + 1 u ) = 3 and ℓ ( t i , j u ) = 2 otherwise, then A Γ , ℓ is the braid group B n . If ℓ ( e ) = 2, for all e P E , then A Γ = x v P V | vw = wv if t v , w u P E y . is the right-angled Artin group associated to Γ . Γ – Γ 1 ô A Γ – A Γ 1 [Kim–Makar-Limanov–Neggers–Roush 80 / Droms 87] A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 9 / 26
C AST OF CHARACTERS A RTIN GROUPS The corresponding Coxeter group , W Γ , ℓ = A Γ , ℓ / x v 2 = 1 | v P V y , � P Γ , ℓ � A Γ , ℓ � W Γ , ℓ � 1 . fits into exact sequence 1 T HEOREM (B RIESKORN 1971) If W Γ , ℓ is finite, then G Γ , ℓ is quasi-projective. Idea: let A Γ , ℓ = reflection arrangement of type W Γ , ℓ (over C ) X Γ , ℓ = C n z Ť H P A Γ , ℓ H , where n = | A Γ , ℓ | P Γ , ℓ = π 1 ( X Γ , ℓ ) then: A Γ , ℓ = π 1 ( X Γ , ℓ / W Γ , ℓ ) = π 1 ( C n zt δ Γ , ℓ = 0 u ) T HEOREM (K APOVICH –M ILLSON 1998) There exist infinitely many ( Γ , ℓ ) such that A Γ , ℓ is not quasi-projective. A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 10 / 26
C OMPARING CLASSES OF GROUPS K ÄHLER GROUPS VS OTHER GROUPS K ÄHLER GROUPS VS OTHER GROUPS Q UESTION (D ONALDSON –G OLDMAN 1989) Which 3-manifold groups are Kähler groups? T HEOREM (D IMCA –S. 2009) Let π be the fundamental group of a closed 3 -manifold. Then π is a ñ π is a finite subgroup of O ( 4 ) , acting freely on S 3 . Kähler group ð Alternative proofs: Kotschick (2012), Biswas, Mj, and Seshadri (2012). T HEOREM (F RIEDL –S. 2014) Let N be a 3 -manifold with non-empty, toroidal boundary. If π 1 ( N ) is a Kähler group, then N – S 1 ˆ S 1 ˆ I. Generalization by Kotschick: If π 1 ( N ) is an infinite Kähler group, then π 1 ( N ) is a surface group. A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 11 / 26
C OMPARING CLASSES OF GROUPS K ÄHLER GROUPS VS OTHER GROUPS T HEOREM (D IMCA –P APADIMA –S. 2009) Let Γ be a finite simple graph, and A Γ the corresponding RAAG. The following are equivalent: A Γ is a Kähler group. 1 A Γ is a free abelian group of even rank. 2 Γ is a complete graph on an even number of vertices. 3 T HEOREM (S. 2011) Let A be an arrangement of lines in CP 2 , with group π = π 1 ( U ( A )) . The following are equivalent: π is a Kähler group. 1 π is a free abelian group of even rank. 2 A consists of an odd number of lines in general position. 3 A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 12 / 26
C OMPARING CLASSES OF GROUPS Q UASI - PROJECTIVE GROUPS VS OTHER GROUPS Q UASI - PROJECTIVE GROUPS VS OTHER GROUPS T HEOREM (D IMCA –P APADIMA –S. 2011) Let π be the fundamental group of a closed, orientable 3 -manifold. Assume π is 1 -formal. Then the following are equivalent: m ( π ) – m ( π 1 ( X )) , for some quasi-projective manifold X. 1 m ( π ) – m ( π 1 ( N )) , where N is either S 3 , # n S 1 ˆ S 2 , or S 1 ˆ Σ g . 2 T HEOREM (F RIEDL –S. 2014) Let N be a 3 -mfd with empty or toroidal boundary. If π 1 ( N ) is a quasi- projective group, then all prime components of N are graph manifolds. In particular, the fundamental group of a hyperbolic 3-manifold with empty or toroidal boundary is never a qp-group. A LEX S UCIU (N ORTHEASTERN ) F UNDAMENTAL GROUPS IN AG & GT F RIBOURG , J UNE 2016 13 / 26
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