R ESONANCE , REPRESENTATIONS , AND THE J OHNSON FILTRATION Alex Suciu Northeastern University, visiting the University of Chicago Geometry/Topology Seminar University of Chicago May 8, 2015 A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 1 / 31
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, 2015) A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 2 / 31
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 ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 3 / 31
T HE J OHNSON FILTRATION F ILTRATIONS AND GRADED L IE ALGEBRAS Let π be a group, with commutator ( x , y ) = xyx ´ 1 y ´ 1 . Suppose given a descending filtration π = Φ 1 Ě Φ 2 Ě ¨ ¨ ¨ Ě Φ s Ě ¨ ¨ ¨ by subgroups of π , satisfying ( Φ s , Φ t ) Ď Φ s + t , @ s , t ě 1 . Then Φ s Ÿ π , and Φ s / Φ s + 1 is abelian. Set à Φ s / Φ s + 1 . gr Φ ( π ) = 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 ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 4 / 31
T HE J OHNSON FILTRATION Basic example: the lower central series , Γ s = Γ s ( π ) , defined as Γ 1 = π , Γ 2 = π 1 , . . . , Γ s + 1 = ( Γ s , π ) , . . . Then for any filtration Φ as above, Γ s Ď Φ s . Thus, we have a morphism of graded Lie algebras, � gr Φ ( π ) . ι Φ : gr Γ ( π ) 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 ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 5 / 31
T HE J OHNSON FILTRATION A UTOMORPHISM GROUPS Let Aut ( π ) be the group of all automorphisms α : π Ñ π , with α ¨ β : = α ˝ β . The Andreadakis–Johnson filtration , Aut ( π ) = F 0 Ě F 1 Ě ¨ ¨ ¨ Ě F s Ě ¨ ¨ ¨ has terms F s = F s ( Aut ( π )) consisting of those automorphisms which act as the identity on the s -th nilpotent quotient of π : F s = ker Aut ( π ) Ñ Aut ( π / Γ s + 1 � � = t α P Aut ( π ) | α ( x ) ¨ x ´ 1 P Γ s + 1 , @ x P π u ( F s , F t ) Ď F s + t . Kaloujnine [1950]: First term is the Torelli group , I π = F 1 = ker � � Aut ( π ) Ñ Aut ( π ab ) . A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 6 / 31
T HE J OHNSON FILTRATION By construction, F 1 = I G is a normal subgroup of F 0 = Aut ( π ) . The quotient group, A ( π ) = F 0 / F 1 = im ( Aut ( π ) Ñ Aut ( π ab )) is the symmetry group of I π ; it fits into the exact sequence � I π � Aut ( π ) � A ( π ) � 1 . 1 The Torelli group comes endowed with two filtrations: The Johnson filtration t F s ( I π ) u s ě 1 , inherited from Aut ( π ) . The lower central series filtration, t Γ s ( I π ) u . The respective associated graded Lie algebras, gr F ( I π ) and gr Γ ( I π ) , come endowed with natural actions of A ( π ) ; moreover, the morphism ι F : gr Γ ( I π ) Ñ gr F ( I π ) is A ( π ) -equivariant. A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 7 / 31
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 π , there is a monomorphism of graded Lie algebras, � Der ( gr Γ ( π )) , J : gr F ( I π ) given on homogeneous elements α P F s ( I π ) and x P Γ t ( π ) by x ) = α ( x ) ¨ x ´ 1 . J ( ¯ α )( ¯ Moreover, J is equivariant with respect to the natural actions of A ( π ) . A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 8 / 31
T HE J OHNSON FILTRATION The Johnson homomorphism informs on the Johnson filtration. T HEOREM Suppose Z ( gr Γ ( π )) = 0 . For each q ě 1 , the following are equivalent: Γ ( I π ) Ñ Der s ( gr Γ ( π )) is injective, for all s ď q. J ˝ ι F : gr s 1 Γ s ( I π ) = F s ( I π ) , for all s ď q + 1 . 2 In particular, if gr Γ ( π ) is centerless and Γ ( I π ) Ñ Der 1 ( gr Γ ( π )) is injective, then F 2 ( I π ) = I 1 J ˝ ι F : gr 1 π . P ROBLEM Determine the homological finiteness properties of the groups F s ( I π ) . In particular, decide whether dim H 1 ( I 1 π , Q ) ă 8 . A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 9 / 31
T HE J OHNSON FILTRATION A N OUTER VERSION Let Inn ( π ) = im ( Ad : π Ñ Aut ( π )) , where Ad x : π Ñ π , y ÞÑ xyx ´ 1 . Define the outer automorphism group of π by q � Inn ( π ) � Aut ( π ) � Out ( π ) � 1 . 1 We then have Filtration t r F s : = q ( F s ) . r F s u s ě 0 on Out ( π ) : F 1 of Out ( π ) . subgroup r I π = r The outer Torelli group of π : � r � Out ( π ) � A ( π ) � 1 . Exact sequence: 1 I π T HEOREM Suppose Z ( gr Γ ( π )) = 0 . Then the Johnson homomorphism induces an A ( π ) -equivariant monomorphism of graded Lie algebras, r F ( r � Ą J : gr r I π ) Der ( gr Γ ( π )) , where Ą Der ( g ) = Der ( g ) / im ( ad : g Ñ Der ( g )) . A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 10 / 31
A LEXANDER INVARIANTS T HE A LEXANDER INVARIANT Let π be a group, and π ab = π / π 1 its maximal abelian quotient. Let π 2 = ( π 1 , π 1 ) ; then π / π 2 is the maximal metabelian quotient. � π 1 / π 2 � π / π 2 � π ab � 0 . Get exact sequence 0 Conjugation in π / π 2 turns the abelian group B ( π ) : = π 1 / π 2 = H 1 ( π 1 , Z ) into a module over R = Z π ab , called the Alexander invariant of π . Since both π 1 and π 2 are characteristic subgroups of π , the action of Aut ( π ) on π induces an action on B ( π ) . This action need not respect the R -module structure. Nevertheless: P ROPOSITION The Torelli group I π acts R-linearly on the Alexander invariant B ( π ) . A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 11 / 31
A LEXANDER INVARIANTS C HARACTERISTIC VARIETIES Assume now that π is finitely generated. π = Hom ( π , C ˚ ) be its character group : an algebraic group, Let p with coordinate ring C [ π ab ] . » The map ab : π ։ π ab induces an isomorphism p Ý Ñ p π . π ab π ˝ – ( C ˚ ) n , where n = rank π ab . p D EFINITION The (first) characteristic variety of π is the support of the (complexified) Alexander invariant B = B ( π ) b C : V ( π ) : = V ( ann B ) Ă p π . This variety informs on the Betti numbers of normal subgroups N Ÿ π with π / N abelian. In particular (for N = π 1 ): P ROPOSITION The set V ( π ) is finite if and only if b 1 ( π 1 ) = dim C B ( π ) b C is finite. A LEX S UCIU (N ORTHEASTERN ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 12 / 31
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 ˚ = ( 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 ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 13 / 31
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 ) R ESONANCE , REPRESENTATIONS & J OHNSON U. C HICAGO , M AY 2015 14 / 31
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