R ESIDUAL FINITENESS PROPERTIES OF FUNDAMENTAL GROUPS Alex Suciu Northeastern University Joint work with Thomas Koberda (U. Virginia) arxiv:1604.02010 Number Theory and Algebraic Geometry Seminar Katholieke Universiteit Leuven May 18, 2016 A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 1 / 23
O UTLINE F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY 1 Fundamental groups of manifolds Projective groups Quasi-projective groups Complements of hypersurfaces Line arrangements R ESIDUALLY FINITE RATIONALLY p GROUPS 2 The RFR p property Characteristic varieties BNS invariants The RFR p topology B OUNDARY MANIFOLDS 3 3-manifolds and the RFR p property Boundary manifolds of curves The RFR p property for boundary manifolds A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 2 / 23
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 ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 3 / 23
F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY P ROJECTIVE GROUPS P ROJECTIVE GROUPS A group π is said to be a projective group if π = π 1 ( M ) , for some smooth, projective variety M . The class of projective groups is closed under finite direct products and passing to finite-index subgroups. [Serre „ 1955] Every finite group is a projective group. The projectivity 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] π cannot split non-trivially as a free product. [Gromov 1989] π = π 1 ( N ) for some closed 3-manifold N iff π is a finite subgroup of O ( 4 ) . [Dimca–S. 2009] A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 4 / 23
F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY 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 ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 5 / 23
F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY 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 . By the Lefschetz hyperplane sections theorem, this class coincides the class of fundamental groups of complements of plane algebraic curves. All such groups are 1-formal. Even more special are the arrangement groups , i.e., the fundamental groups of complements of complex hyperplane arrangements (or, equivalently, complex line arrangements). A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 6 / 23
F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY C OMPLEMENTS OF HYPERSURFACES P LANE ALGEBRAIC CURVES Let C Ă CP 2 be a plane algebraic curve, defined by a homogeneous polynomial f P C [ z 1 , z 2 , z 3 ] . Zariski commissioned Van Kampen to find a presentation for the fundamental group of the complement, U ( C ) = CP 2 z C . Using the Alexander polynomial, Zariski showed that π = π 1 ( U ) is not fully determined by the combinatorics of C , but depends on the position of its singularities. P ROBLEM (Z ARISKI ) Is π residually finite ? That is, given g P π , g ‰ 1, is there is a homomorphism ϕ : π Ñ G onto some finite group G such that ϕ ( g ) ‰ 1. Equivalently, is the canonical morphism to the profinite completion π Ñ π alg : = lim Ý N Ÿ π : [ π : N ] ă8 π / N , injective? Ð A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 7 / 23
F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY L INE ARRANGEMENTS L INE 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 L 1 L 2 L 3 L 4 P 1 P 2 A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 8 / 23
F UNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY L INE ARRANGEMENTS 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 LCS 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 ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 9 / 23
R ESIDUALLY FINITE RATIONALLY p GROUPS T HE RFR p PROPERTY T HE RFR p PROPERTY Let G be a finitely generated group and let p be a prime. We say that G is residually finite rationally p if there exists a sequence of subgroups G = G 0 ą ¨ ¨ ¨ ą G i ą G i + 1 ą ¨ ¨ ¨ such that G i + 1 Ÿ G i . 1 Ş i ě 0 G i = t 1 u . 2 G i / G i + 1 is an elementary abelian p -group. 3 ker ( G i Ñ H 1 ( G i , Q )) ă G i + 1 . 4 Remarks: We may assume that each G i Ÿ G . G is RFR p if and only if rad p ( G ) : = Ş i G i is trivial. For each prime p , there exists a finitely presented group G p which is RFR p , but not RFR q for any prime q ‰ p . A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 10 / 23
R ESIDUALLY FINITE RATIONALLY p GROUPS T HE RFR p PROPERTY G RFR p ñ residually p ñ residually finite and residually nilpotent. G RFR p ñ G torsion-free. G finitely presented and RFR p ñ G has solvable word problem. The class of RFR p groups is closed under these operations: Taking subgroups. Finite direct products. Finite free products. The following groups are RFR p , for all p : Finitely generated free groups. Closed, orientable surface groups. Right-angled Artin groups. The following groups are not RFR p , for any p : Finite groups Non-abelian nilpotent groups A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 11 / 23
R ESIDUALLY FINITE RATIONALLY p GROUPS C HARACTERISTIC VARIETIES Let G be a finitely-generated group, and let p G = Hom ( G , C ˚ ) . The (degree 1) characteristic varieties of G are the closed algebraic subsets V i ( G ) = t χ P p G | dim H 1 ( G , C χ ) ě i u . L EMMA Let G 2 = [ G 1 , G 1 ] . The projection map π : G Ñ G / G 2 induces an π : { G / G 2 Ñ p isomorphism ˆ G which restricts to isomorphisms V i ( G / G 2 ) Ñ V i ( G ) for all i ě 1 . A group G is large if G virtually surjects onto a non-abelian free group. L EMMA (K OBERDA 2014) An f.p. group G is large if and only if there exists a finite-index subgroup H ă G such that V 1 ( H ) has infinitely many torsion points. A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 12 / 23
R ESIDUALLY FINITE RATIONALLY p GROUPS C HARACTERISTIC VARIETIES T HEOREM Let G be a non-abelian, finitely generated group which is RFRp for infinitely many primes. Then: G / G 2 is not finitely presented. G 1 is not finitely generated. V 1 ( G ) contains infinitely many torsion points. As a consequence, we obtain the following ‘Tits alternative’ for RFR p groups. C OROLLARY Let G be a finitely presented group which is RFRp for infinitely many primes. Then either: G is abelian. 1 G is large. 2 A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 13 / 23
R ESIDUALLY FINITE RATIONALLY p GROUPS BNS INVARIANTS BNS INVARIANTS The Bieri–Neumann–Strebel invariant of a f.g. group G is the set Σ 1 ( G ) = t χ P S ( G ) | Cay χ ( G ) is connected u , where S ( G ) is the unit sphere in H 1 ( G , R ) . For each non-zero homomorphism χ : G Ñ R , we let Cay χ ( G ) be the induced subgraph on vertices g P G such that χ ( g ) ě 0. Although Cay χ ( G ) depends on the choice of a (finite, symmetric) generating set for G , its connectivity is independent of such choice. T HEOREM (P APADIMA –S. (2010)) � A . Σ 1 ( G ) Ă τ R � 1 ( V 1 ( G )) Here, if V Ă ( C ˚ ) n , then τ 1 ( V ) = t z P C n | exp ( λ z ) P V , @ λ P C u . A LEX S UCIU (N ORTHEASTERN ) R ESIDUAL FINITENESS PROPERTIES KU L EUVEN , M AY 2016 14 / 23
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