A UTOMORPHISM GROUPS , L IE ALGEBRAS , AND RESONANCE VARIETIES Alex Suciu Northeastern University Colloquium University of Western Ontario April 25, 2013 A LEX S UCIU (N ORTHEASTERN ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 1 / 30
R EFERENCES Stefan Papadima and Alexander I. Suciu, Homological finiteness in the Johnson filtration of the automorphism group of a free group , Journal of Topology 5 (2012), no. 4, 909–944. Stefan Papadima and Alexander I. Suciu, Vanishing resonance and representations of Lie algebras , arxiv:1207.2038 A LEX S UCIU (N ORTHEASTERN ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 2 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 3 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 4 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 5 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 6 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 7 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 8 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 9 / 30
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 � A ( G ) � Out ( 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 T G ) Der ( gr Γ ( G )) , J : gr r where Ą Der ( g ) = Der ( g ) / im ( ad : g Ñ Der ( g )) . A LEX S UCIU (N ORTHEASTERN ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 10 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 11 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 12 / 30
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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 13 / 30
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 j ^ ¨ ¨ ¨ ^ v i p ) . v i 1 ^ ¨ ¨ ¨ ^ v i p ÞÑ 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 � � B ( V , K ) . � S b C � 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 ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 14 / 30
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 0 � A 1 � A 2 is not exact (in the middle). Using work of R. Fröberg and C. Löfwall on Koszul homology, the graded pieces of the (dual) Koszul module can be reinterpreted in terms of the linear strand in an appropriate Tor module: q – Tor E B ˚ q + 1 ( A , C ) q + 2 A LEX S UCIU (N ORTHEASTERN ) G ROUPS , L IE ALGEBRAS , AND RESONANCE UWO, A PRIL 2013 15 / 30
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