: gr ( G ) E XAMPLE (P. H ALL , E. W ITT , W. M AGNUS ) Let F n = - PowerPoint PPT Presentation

H OMOLOGICAL FINITENESS IN THE J OHNSON FILTRATION Alex Suciu Northeastern University Geometry & Topology Seminar Yale University February 4, 2014 A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 1 / 32

R EFERENCES Stefan Papadima and Alexander I. Suciu Homological finiteness in the Johnson filtration of the automorphism group of a free group J. Topol. 5 (2012), no. 4, 909–944. Vanishing resonance and representations of Lie algebras J. Reine Angew. Math. (to appear) A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 2 / 32

O UTLINE T HE J OHNSON FILTRATION 1 A LEXANDER INVARIANTS 2 R ESONANCE VARIETIES 3 R OOTS , WEIGHTS , AND VANISHING RESONANCE 4 A UTOMORPHISM GROUPS OF FREE GROUPS 5 A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 3 / 32

T HE J OHNSON FILTRATION F ILTRATIONS AND GRADED L IE ALGEBRAS Let G be a group, with commutator ( x , y ) = xyx ´ 1 y ´ 1 . Suppose given a descending filtration G = Φ 1 Ě Φ 2 Ě ¨ ¨ ¨ Ě Φ s Ě ¨ ¨ ¨ by subgroups of G , satisfying ( Φ s , Φ t ) Ď Φ s + t , @ s , t ě 1 . Then Φ s Ÿ G , and Φ s / Φ s + 1 is abelian. Set à Φ s / Φ s + 1 . gr Φ ( G ) = s ě 1 Φ Ñ gr s + t This is a graded Lie algebra, with bracket [ , ] : gr s Φ ˆ gr t Φ induced by the group commutator. A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 4 / 32

T HE J OHNSON FILTRATION Basic example: the lower central series , Γ s = Γ s ( G ) , defined as Γ 1 = G , Γ 2 = G 1 , . . . , Γ s + 1 = ( Γ s , G ) , . . . Then for any filtration Φ as above, Γ s Ď Φ s ; thus, we have a morphism of graded Lie algebras, � gr Φ ( G ) . ι Φ : gr Γ ( G ) E XAMPLE (P. H ALL , E. W ITT , W. M AGNUS ) Let F n = x x 1 , . . . , x n y be the free group of rank n . Then: F n is residually nilpotent, i.e., Ş s ě 1 Γ s ( F n ) = t 1 u . gr Γ ( F n ) is isomorphic to the free Lie algebra L n = Lie ( Z n ) . ř s gr s Γ ( F n ) is free abelian, of rank 1 d . d | s µ ( d ) n s If n ě 2, the center of L n is trivial. A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 5 / 32

T HE J OHNSON FILTRATION A UTOMORPHISM GROUPS Let Aut ( G ) be the group of all automorphisms α : G Ñ G , with α ¨ β : = α ˝ β . The Andreadakis–Johnson filtration , Aut ( G ) = F 0 Ě F 1 Ě ¨ ¨ ¨ Ě F s Ě ¨ ¨ ¨ has terms F s = F s ( Aut ( G )) consisting of those automorphisms which act as the identity on the s -th nilpotent quotient of G : F s = ker Aut ( G ) Ñ Aut ( G / Γ s + 1 � � = t α P Aut ( G ) | α ( x ) ¨ x ´ 1 P Γ s + 1 , @ x P G u ( F s , F t ) Ď F s + t . Kaloujnine [1950]: First term is the Torelli group , T G = F 1 = ker � � Aut ( G ) Ñ Aut ( G ab ) . A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 6 / 32

T HE J OHNSON FILTRATION By construction, F 1 = T G is a normal subgroup of F 0 = Aut ( G ) . The quotient group, A ( G ) = F 0 / F 1 = im ( Aut ( G ) Ñ Aut ( G ab )) is the symmetry group of T G ; it fits into exact sequence � T G � Aut ( G ) � A ( G ) � 1 . 1 The Torelli group comes endowed with two filtrations: The Johnson filtration t F s ( T G ) u s ě 1 , inherited from Aut ( G ) . The lower central series filtration, t Γ s ( T G ) u . The respective associated graded Lie algebras, gr F ( T G ) and gr Γ ( T G ) , come endowed with natural actions of A ( G ) ; moreover, the morphism ι F : gr Γ ( T G ) Ñ gr F ( T G ) is A ( G ) -equivariant. A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 7 / 32

T HE J OHNSON FILTRATION T HE J OHNSON HOMOMORPHISM Given a graded Lie algebra g , let Der s ( g ) = t δ : g ‚ Ñ g ‚ + s linear | δ [ x , y ] = [ δ x , y ] + [ x , δ y ] , @ x , y P g u . Then Der ( g ) = À s ě 1 Der s ( g ) is a graded Lie algebra, with bracket [ δ , δ 1 ] = δ ˝ δ 1 ´ δ 1 ˝ δ . T HEOREM Given a group G, there is a monomorphism of graded Lie algebras, � Der ( gr Γ ( G )) , J : gr F ( T G ) given on homogeneous elements α P F s ( T G ) and x P Γ t ( G ) by x ) = α ( x ) ¨ x ´ 1 . J ( ¯ α )( ¯ Moreover, J is equivariant with respect to the natural actions of A ( G ) . A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 8 / 32

T HE J OHNSON FILTRATION The Johnson homomorphism informs on the Johnson filtration. T HEOREM Let G be a group. For each q ě 1 , the following are equivalent: Γ ( T G ) Ñ Der s ( gr Γ ( G )) is injective, for all s ď q. J ˝ ι F : gr s 1 Γ s ( T G ) = F s ( T G ) , for all s ď q + 1 . 2 P ROPOSITION Suppose G is residually nilpotent, gr Γ ( G ) is centerless, and Γ ( T G ) Ñ Der 1 ( gr Γ ( G )) is injective. Then F 2 ( T G ) = T 1 J ˝ ι F : gr 1 G . P ROBLEM Determine the homological finiteness properties of the groups F s ( T G ) . In particular, decide whether dim H 1 ( T 1 G , Q ) ă 8 . A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 9 / 32

T HE J OHNSON FILTRATION A N OUTER VERSION Let Inn ( G ) = im ( Ad : G Ñ Aut ( G )) , where Ad x : G Ñ G , y ÞÑ xyx ´ 1 . Define the outer automorphism group of a group G by π � Inn ( G ) � Aut ( G ) � Out ( G ) � 1 . 1 We then have Filtration t r F s : = π ( F s ) . r F s u s ě 0 on Out ( G ) : F 1 of Out ( G ) . subgroup r T G = r The outer Torelli group of G : � r � Out ( G ) � A ( G ) � 1 . Exact sequence: 1 T G T HEOREM Suppose Z ( gr Γ ( G )) = 0 . Then the Johnson homomorphism induces an A ( G ) -equivariant monomorphism of graded Lie algebras, r F ( r � Ą J : gr r T G ) Der ( gr Γ ( G )) , where Ą Der ( g ) = Der ( g ) / im ( ad : g Ñ Der ( g )) . A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 10 / 32

A LEXANDER INVARIANTS T HE A LEXANDER INVARIANT Let G be a group, and G ab = G / G 1 its maximal abelian quotient. Let G 2 = ( G 1 , G 1 ) ; then G / G 2 is the maximal metabelian quotient. � G 1 / G 2 � G / G 2 � G ab � 0 . Get exact sequence 0 Conjugation in G / G 2 turns the abelian group B ( G ) : = G 1 / G 2 = H 1 ( G 1 , Z ) into a module over R = Z G ab , called the Alexander invariant of G . Since both G 1 and G 2 are characteristic subgroups of G , the action of Aut ( G ) on G induces an action on B ( G ) . This action need not respect the R -module structure. Nevertheless: P ROPOSITION The Torelli group T G acts R-linearly on the Alexander invariant B ( G ) . A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 11 / 32

A LEXANDER INVARIANTS C HARACTERISTIC VARIETIES Let G be a finitely generated group. Let p G = Hom ( G , C ˚ ) be its character group : an algebraic group, with coordinate ring C [ G ab ] . The map ab : G ։ G ab induces an isomorphism p » Ñ p G ab Ý G . G ˝ – ( C ˚ ) n , where n = rank G ab . p D EFINITION The (first) characteristic variety of G is the support of the (complexified) Alexander invariant B = B ( G ) b C : V ( G ) : = V ( ann B ) Ă p G . This variety informs on the Betti numbers of normal subgroups H Ÿ G with G / H abelian. In particular (for H = G 1 ): P ROPOSITION The set V ( G ) is finite if and only if b 1 ( G 1 ) = dim C B ( G ) b C is finite. A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 12 / 32

R ESONANCE VARIETIES R ESONANCE VARIETIES Let V be a finite-dimensional C -vector space, and let K Ă V ^ V be a subspace. D EFINITION The resonance variety R = R ( V , K ) is the set of elements a P V ˚ for which there is an element b P V ˚ , not proportional to a , such that a ^ b belongs to the orthogonal complement K K Ď V ˚ ^ V ˚ . R is a conical, Zariski-closed subset of the affine space V ˚ . For instance, if K = 0 and dim V ą 1, then R = V ˚ . At the other extreme, if K = V ^ V , then R = 0. The resonance variety R has several other interpretations. A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 13 / 32

R ESONANCE VARIETIES K OSZUL MODULES Let S = Sym ( V ) be the symmetric algebra on V . Ź V , δ ) be the Koszul resolution, with differential Let ( S b C Ź p V Ñ S b C Ź p ´ 1 V given by δ p : S b C ÿ p j = 1 ( ´ 1 ) j ´ 1 v i j b ( v i 1 ^ ¨ ¨ ¨ ^ p v i 1 ^ ¨ ¨ ¨ ^ v i p ÞÑ v i j ^ ¨ ¨ ¨ ^ v i p ) . Let ι : K Ñ V ^ V be the inclusion map. The Koszul module B ( V , K ) is the graded S -module presented as � Ź 3 V ‘ K Ź 2 V δ 3 + id b ι � S b C � � B ( V , K ) . � S b C P ROPOSITION The resonance variety R = R ( V , K ) is the support of the Koszul module B = B ( V , K ) : R = V ( ann ( B )) Ă V ˚ . In particular, R = 0 if and only if dim C B ă 8 . A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 14 / 32

R ESONANCE VARIETIES C OHOMOLOGY JUMP LOCI Let A = A ( V , K ) be the quadratic algebra defined as the quotient of the exterior algebra E = Ź V ˚ by the ideal generated by K K Ă V ˚ ^ V ˚ = E 2 . Then R is the set of points a P A 1 where the cochain complex a a � A 1 � A 2 A 0 is not exact (in the middle). The graded pieces of the (dual) Koszul module can be reinterpreted in terms of the linear strand in a Tor module: q – Tor E B ˚ q + 1 ( A , C ) q + 2 A LEX S UCIU (N ORTHEASTERN ) H OMOLOGY IN THE J OHNSON FILTRATION Y ALE , F EB 2014 15 / 32