Alex Suciu Northeastern University Workshop on Geometric Group - PowerPoint PPT Presentation

C OMPLEX GEOMETRY AND 3- DIMENSIONAL TOPOLOGY Alex Suciu Northeastern University Workshop on Geometric Group Theory and Geometric Topology University of Virginia October 17, 2015 A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 1 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY 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 ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 2 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY K ÄHLER GROUPS & 3- MANIFOLD GROUPS K ÄHLER GROUPS & 3- MANIFOLD 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. Examples: smooth, complex projective varieties. If M is a Kähler manifold, π = π 1 ( M ) is called a Kähler group . This also puts strong restrictions on π , e.g.: b 1 ( π ) is even (Hodge theory) π is 1-formal: Malcev Lie algebra m ( π ) is quadratic (DGMS 1975) π cannot split non-trivially as a free product (Gromov 1989) π finite ñ π projective group (Serre 1958). Q UESTION (D ONALDSON –G OLDMAN 1989) Which 3-manifold groups are Kähler groups? Reznikov (2002) gave a partial solution. A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 3 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY K ÄHLER GROUPS & 3- MANIFOLD 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 have since been given by Kotschick (2012) and by Biswas, Mj and Seshadri (2012). T HEOREM (F RIEDL –S. 2013) 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. Since then, Kotschick has generalized this result, by dropping the toroidal boundary assumption: T HEOREM (K OTSCHICK 2013) If π 1 ( N ) is an infinite Kähler group, then π 1 ( N ) is a surface group. A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 4 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY Q UASI - PROJECTIVE GROUPS & 3- MANIFOLD GROUPS Q UASI - PROJECTIVE GROUPS & 3- MANIFOLD GROUPS A group π is called 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. Subclass: fundamental groups of complements of hypersurfaces in CP n , or, equivalently, fundamental groups of complements of plane algebraic curves. Such groups are 1-formal. A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 5 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY Q UASI - PROJECTIVE GROUPS & 3- MANIFOLD GROUPS Q UESTION (D IMCA –S. 2009) Which 3-manifold groups are quasi-projective 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 A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 6 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY Q UASI - PROJECTIVE GROUPS & 3- MANIFOLD GROUPS Joint work with Stefan Friedl (2013) T HEOREM 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 ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 7 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY A LEXANDER POLYNOMIALS A LEXANDER POLYNOMIALS Let H be a finitely generated, free abelian group. Let M be a finitely generated module over Λ = Z [ H ] . Pick a α � Λ s � M � 0 with p ě s . presentation Λ p Let E k ( M ) be the ideal of minors of size s ´ k of α , and set ord k ( M ) : = gcd ( E k ( M )) P Λ (well-defined up to units in Λ ). M = Λ r ‘ Tors ( M ) and set ∆ r M : = ord 0 ( Tors M ) . Define the thickness of M as th ( M ) = dim Newt ( ∆ r M ) . A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 8 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY A LEXANDER POLYNOMIALS Let X be a finite, conn. CW-complex. Write H : = H 1 ( X ; Z ) / Tors. Alexander invariant: A X = H 1 ( X ; Z [ H ]) . Alexander polynomials: ∆ k X = ord k ( A X ) ; usual one: ∆ = ∆ 0 . Set th ( X ) : = th ( A X ) . Note: th ( X ) = th ( π 1 ( X )) . Let p H = Hom ( H , C ˚ ) be the character torus. Define hypersurfaces X ) = t ρ P p V ( ∆ k H | ∆ k X ( ρ ) = 0 u . If X = S 3 z K , then ∆ X is the classical Alexander polynomial of K , X ) Ă C ˚ is the set of roots of ∆ X , of multiplicity at least k . and V ( ∆ k Also define the (degree 1) characteristic varieties of X as V k ( X ) = t ρ P p H | dim H 1 ( X , C ρ ) ě k u , where C ρ = C , viewed as a module over Z H , via g ¨ x = ρ ( g ) x . We then have: V k ( X ) zt 1 u = V ( E k ´ 1 ( A X )) zt 1 u . A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 9 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY A LEXANDER POLYNOMIALS Let ˇ V k ( X ) be the union of all codim 1 irreducible components of V k ( X ) . L EMMA (DPS08 FOR k = 0 , FS13 FOR k ą 0 ) H, in which case ˇ ∆ k ´ 1 = 0 if and only if V k ( X ) = p V k ( X ) = H . 1 X Suppose b 1 ( X ) ě 1 and ∆ k ´ 1 ‰ 0 . Then at least away from 1 , 2 X ˇ V k ( X ) = V ( ∆ k ´ 1 ) . X T HEOREM (DPS, FS) . = const if and only if ˇ Suppose b 1 ( X ) ě 2 . Then ∆ k ´ 1 V k ( X ) = H . X Otherwise, the following are equivalent: The Newton polytope of ∆ k ´ 1 is a line segment. 1 X All irreducible components of ˇ V k ( X ) are parallel, codim 1 subtori 2 of p H. A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 10 / 30

F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY A LEXANDER POLYNOMIALS The next theorem is due to Arapura (1997), with improvements by DPS (2008, 2009) and Artal-Bartolo, Cogolludo, Matei (2010). T HEOREM Let π be a quasi-projective group. Then, for each k ě 1 , The irreducible components of V k ( π ) are (possibly torsion-translated) subtori of the character torus p H. Any two distinct components of V k ( π ) meet in a finite set. Using this theorem, we prove T HEOREM (DPS08 FOR k = 0 , FS13 FOR k ą 0 ) Let π be a quasi-projective group, and assume b 1 ( π ) ‰ 2 . Then, for each k ě 0 , the polynomial ∆ k π is either zero, or the Newton polytope of ∆ k π is a point or a line segment. In particular, th ( π ) ď 1 . A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 11 / 30

3- MANIFOLD GROUPS T HURSTON NORM AND A LEXANDER NORM T HURSTON NORM AND A LEXANDER NORM Let N be a 3-manifold with either empty or toroidal boundary. A class φ P H 1 ( N ; Z ) = Hom ( π 1 ( N ) , Z ) is fibered if there exists a fibration p : N Ñ S 1 such that p ˚ : π 1 ( N ) Ñ Z coincides with φ . Given a surface Σ with connected components Σ 1 , . . . , Σ s , put χ ´ ( Σ ) = ř s i = 1 max t´ χ ( Σ i ) , 0 u . � Thurston norm : } φ } T = min χ ´ ( Σ ) u , where Σ runs through all the properly embedded surfaces dual to φ . } ´ } T defines a (semi)norm on H 1 ( N ; Z ) , which can be extended to a (semi)norm } ´ } T on H 1 ( N ; Q ) . The unit norm ball, B T = t φ P H 1 ( N ; Q ) | } φ } T ď 1 u , is a rational polyhedron with finitely many sides, symmetric in the origin. A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 12 / 30

3- MANIFOLD GROUPS T HURSTON NORM AND A LEXANDER NORM The set of fibered classes form a cone on certain open, top-dimensional faces of B T , called the fibered faces of B T . Two faces F and G are equivalent if F = ˘ G . Clearly, F is fibered if and only if ´ F is fibered. We say φ P H 1 ( N ; Q ) is quasi-fibered if it lies on the boundary of a fibered face of B T . Results of Stallings (1962) and Gabai (1983) imply C OROLLARY (FS13) Let p : N 1 Ñ N be a finite cover. Then: φ P H 1 ( N ; Q ) quasi-fibered ñ p ˚ ( φ ) P H 1 ( N 1 ; Q ) quasi-fibered. 1 Pull-backs of inequivalent faces of the Thurston norm ball of N lie 2 on inequivalent faces of the Thurston norm ball of N 1 . A LEX S UCIU (N ORTHEASTERN ) C OMPLEX GEOMETRY AND 3 D TOPOLOGY V IRGINIA W ORKSHOP 13 / 30