RESONANCE, REPRESENTATIONS, AND THE JOHNSON FILTRATION

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1 RESONANCE, REPRESENTATIONS, AND THE JOHNSON FILTRATION Alex Suciu Northeastern University, visiting the University of Chicago Geometry/Topology Seminar University of Chicago May 8, 2015 ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

2 REFERENCES 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, Vanishing resonance and representations of Lie algebras J. Reine Angew. Math. (to appear, 2015) ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

3 OUTLINE 1 THE JOHNSON FILTRATION 2 ALEXANDER INVARIANTS 3 RESONANCE VARIETIES 4 ROOTS, WEIGHTS, AND VANISHING RESONANCE 5 AUTOMORPHISM GROUPS OF FREE GROUPS ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

4 THE JOHNSON FILTRATION FILTRATIONS AND GRADED LIE ALGEBRAS Let π be a group, with commutator (x, y) = xyx 1 y 1. Suppose given a descending filtration by subgroups of π, satisfying π = Φ 1 Ě Φ 2 Ě Ě Φ s Ě (Φ s, Φ t ) Ď Φ t ě 1. Then Φ s Ÿ π, and Φ s /Φ s+1 is abelian. Set gr Φ (π) = à sě1 Φ s /Φ s+1. This is a graded Lie algebra, with bracket [, ] : gr s Φ ˆ grt Φ Ñ grs+t Φ induced by the group commutator. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

5 THE JOHNSON 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, ι Φ : gr Γ (π) gr Φ (π). EXAMPLE (P. HALL, E. WITT, W. MAGNUS) Let F n = xx 1,..., x n y be the free group of rank n. Then: F n is residually nilpotent, i.e., Ş sě1 Γs (F n ) = t1u. gr Γ (F n ) is isomorphic to the free Lie algebra L n = Lie(Z n ). gr s Γ (F n) is free abelian, of rank 1 ř s d s µ(d)n d s. If n ě 2, the center of L n is trivial. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

6 THE JOHNSON FILTRATION AUTOMORPHISM 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) = tα P Aut(π) α(x) x 1 P Γ P πu Kaloujnine [1950]: (F s, F t ) Ď F s+t. First term is the Torelli group, I π = F 1 = ker ( Aut(π) Ñ Aut(π ab ) ). ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

7 THE JOHNSON 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 1 I π Aut(π) A(π) 1. The Torelli group comes endowed with two filtrations: The Johnson filtration tf s (I π )u sě1, inherited from Aut(π). The lower central series filtration, tγ 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. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

8 THE JOHNSON FILTRATION THE JOHNSON HOMOMORPHISM Given a graded Lie algebra g, let Der s (g) = tδ : g Ñ g +s linear δ[x, y] = [δx, y] + [x, y P gu. Then Der(g) = À sě1 Ders (g) is a graded Lie algebra, with bracket [δ, δ 1 ] = δ δ 1 δ 1 δ. THEOREM Given a group π, there is a monomorphism of graded Lie algebras, J : gr F (I π ) Der(gr Γ (π)), given on homogeneous elements α P F s (I π ) and x P Γ t (π) by J(ᾱ)( x) = α(x) x 1. Moreover, J is equivariant with respect to the natural actions of A(π). ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

9 THE JOHNSON FILTRATION The Johnson homomorphism informs on the Johnson filtration. THEOREM Suppose Z (gr Γ (π)) = 0. For each q ě 1, the following are equivalent: 1 J ι F : gr s Γ (I π) Ñ Der s (gr Γ (π)) is injective, for all s ď q. 2 Γ s (I π ) = F s (I π ), for all s ď q + 1. In particular, if gr Γ (π) is centerless and J ι F : gr 1 Γ (I π) Ñ Der 1 (gr Γ (π)) is injective, then F 2 (I π ) = I 1 π. PROBLEM 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. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

10 THE JOHNSON FILTRATION AN 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 1 Inn(π) Aut(π) q Out(π) 1. We then have Filtration tf rs u sě0 on Out(π): F r s := q(f s ). The outer Torelli group of π: subgroup I r π = F r1 of Out(π). Exact sequence: 1 r I π Out(π) A(π) 1. THEOREM Suppose Z (gr Γ (π)) = 0. Then the Johnson homomorphism induces an A(π)-equivariant monomorphism of graded Lie algebras, rj : gr rf ( r I π ) Ą Der(grΓ (π)), where Ą Der(g) = Der(g)/ im(ad: g Ñ Der(g)). ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

11 ALEXANDER INVARIANTS THE ALEXANDER 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. Get exact sequence 0 π 1 /π 2 π/π 2 π ab 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 = Zπ 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: PROPOSITION The Torelli group I π acts R-linearly on the Alexander invariant B(π). ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

12 ALEXANDER INVARIANTS CHARACTERISTIC VARIETIES Assume now that π is finitely generated. Let pπ = Hom(π, C ) be its character group: an algebraic group, with coordinate ring C[π ab ].» The map ab: π π ab induces an isomorphism pπ ab ÝÑ pπ. pπ (C ) n, where n = rank π ab. DEFINITION 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 pπ. This variety informs on the Betti numbers of normal subgroups N Ÿ π with π/n abelian. In particular (for N = π 1 ): PROPOSITION The set V(π) is finite if and only if b 1 (π 1 ) = dim C B(π) b C is finite. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

13 RESONANCE VARIETIES RESONANCE VARIETIES Let V be a finite-dimensional C-vector space, and let K Ă V ^ V be a subspace. DEFINITION 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. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

14 RESONANCE VARIETIES KOSZUL MODULES Let S = Sym(V ) be the symmetric algebra on V. Let (S b C Ź V, δ) be the Koszul resolution, with differential δ p : S b C Ź p V Ñ S b C Ź p 1 V given by v i1 ^ ^ v ip ÞÑ ÿ p j=1 ( 1)j 1 v ij b (v i1 ^ ^ pv ij ^ ^ v ip ). Let ι : K Ñ V ^ V be the inclusion map. The Koszul module B(V, K ) is the graded S-module presented as S b C ( Ź 3 V K ) δ 3 +id bι S b C Ź 2 V B(V, K ). PROPOSITION 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. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

15 RESONANCE VARIETIES COHOMOLOGY 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 is not exact (in the middle). A 0 a A 1 a A 2 The graded pieces of the (dual) Koszul module can be reinterpreted in terms of the linear strand in a Tor module: B q Tor E q+1(a, C) q+2 ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

16 RESONANCE VARIETIES VANISHING RESONANCE Setting m = dim K, we may view K as a point in Gr m (V ^ V ), and P(K K ) as a codimension m projective subspace in P(V ^ V ). LEMMA Let Gr 2 (V ) ãñ P(V ^ V ) be the Plücker embedding. Then, R(V, K ) = 0 ðñ P(K K ) X Gr 2 (V ) = H. THEOREM For any integer m with 0 ď m ď ( n 2 ), where n = dim V, the set U n,m = K P Gr m (V ^ V ) R(V, K ) = 0 ( is Zariski open. Moreover, this set is non-empty if and only if m ě 2n 3, in which case there is an integer q = q(n, m) such that B q (V, K ) = 0, for every K P U n,m. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

17 RESONANCE VARIETIES RESONANCE VARIETIES OF GROUPS The resonance variety of a f.g. group π is defined as R(π) = R(V, K ), where V = H 1 (π, C) and K K = ker(y π : V ^ V Ñ H 2 (π, C)). Rationally, every resonance variety arises in this fashion: PROPOSITION Let V be a finite-dimensional C-vector space, and let K Ď V ^ V be a linear subspace, defined over Q. Then, there is a finitely presented, commutator-relators group π with V = H 1 (π, C) and K K = ker(y π ). R = R(π) is an approximation to V = V(π). THEOREM (LIBGOBER, DIMCA PAPADIMA S.) Let TC 1 (V) be the tangent cone to V at 1, viewed as a subset of T 1 ( pπ) = H 1 (π, C). Then TC 1 (V) Ď R. Moreover, if π is 1-formal, then equality holds, and R is a union of rational subspaces. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

18 RESONANCE VARIETIES EXAMPLE (RIGHT-ANGLED ARTIN GROUPS) Let Γ = (V, E) be a (finite, simple) graph. The corresponding right-angled Artin group is π Γ = xv P V vw = wv if tv, wu P Ey. V = H 1 (π Γ, C) is the vector space spanned by V. K Ď V ^ V is spanned by tv ^ w tv, wu P Eu. A = A(V, K ) is the exterior Stanley Reisner ring of Γ. R(π Γ ) is the union of all coordinate subspaces C W Ă C V, taken over all W Ă V for which the induced graph Γ W is disconnected. ř qě0 dim C(B q )t q+2 = Q Γ (t/(1 t)), where Q Γ (t) = ř ř b kě0 WĂV: W =k 0 (Γ W )t k. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

19 ROOTS, WEIGHTS, AND VANISHING RESONANCE ROOTS, WEIGHTS, AND VANISHING RESONANCE Let g be a complex, semisimple Lie algebra. Fix a Cartan subalgebra h Ă g and a set of simple roots Ă h. Let (, ) be the inner product on h defined by the Killing form. Each simple root β P gives rise to elements x β, y β P g and h β P h which generate a subalgebra of g isomorphic to sl 2 (C). Each irreducible representation of g is of the form V (λ), where λ P h Q is a dominant weight. A non-zero vector v P V (λ) is a maximal vector (of weight λ) if x β v = 0, for all β P. Such a vector is uniquely determined (up to non-zero scalars), and is denoted by v λ. LEMMA The representation V (λ) ^ V (λ) contains a direct summand isomorphic to V (2λ β), for some simple root β, if and only if (λ, β) 0. When it exists, such a summand is unique. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

20 ROOTS, WEIGHTS, AND VANISHING RESONANCE THEOREM Let V = V (λ) be an irreducible g-module, and let K Ă V ^ V be a submodule. Let V = V (λ ) be the dual module, and let v λ be a maximal vector for V. 1 Suppose there is a root β P such that (λ, β) 0, and suppose the vector v λ ^ y β v λ (of weight 2λ β) belongs to K K. Then R(V, K ) 0. 2 Suppose that 2λ β is not a dominant weight for K K, for any simple root β. Then R(V, K ) = 0. COROLLARY R(V, K ) = 0 if and only if 2λ β is not a dominant weight for K K, for any simple root β such that (λ, β) 0. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

21 ROOTS, WEIGHTS, AND VANISHING RESONANCE THE CASE OF g = sl 2 (C) h is spanned t 1 and t 2 (the dual coordinates on the subspace of diagonal 2 ˆ 2 complex matrices), subject to t 1 + t 2 = 0. There is a single simple root, β = t 1 t 2. The defining representation is V (λ 1 ), where λ 1 = t 1. The irreps are of the form V n = V (nλ 1 ) = Sym n (V (λ 1 )), for some n ě 0. Moreover, dim V n = n + 1 and V n = V n. The second exterior power of V n decomposes into irreducibles, according to the Clebsch-Gordan rule: V n ^ V n = à jě0 V 2n 2 4j. These summands occur with multiplicity 1, and V 2n 2 is always one of those summands. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

22 ROOTS, WEIGHTS, AND VANISHING RESONANCE PROPOSITION Let K be an sl 2 (C)-submodule of V n ^ V n. TFAE: 1 The variety R(V n, K ) consists only of 0 P V n. 2 The C-vector space B(V n, K ) is finite-dimensional. 3 The representation K contains V 2n 2 as a direct summand. The Sym(V n )-modules W (n) = B(V n, V 2n 2 ) were studied by Weyman and Eisenbud (1990). We strengthen one of their results: COROLLARY For any sl 2 (C)-submodule K Ă V n ^ V n, the Koszul module B(V n, K ) is finite-dimensional over C if and only if B(V n, K ) is a quotient of W (n). Open problem: compute Hilb(W (n)). The vanishing of W n 2 (n), for all n ě 1, would imply the generic Green Conjecture on free resolutions of canonical curves. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

23 AUTOMORPHISM GROUPS OF FREE GROUPS AUTOMORPHISM GROUPS OF FREE GROUPS Identify (F n ) ab = Z n, and Aut(Z n ) = GL n (Z). The morphism Aut(F n ) Ñ GL n (Z) is onto; thus, A(F n ) = GL n (Z). Denote the Torelli group by IA n = I Fn, and the Johnson Andreadakis filtration by J s n = F s (Aut(F n )). Magnus [1934]: IA n is generated by the automorphisms # # x i ÞÑ x j x i x 1 j x i ÞÑ x i (x j, x k ) α ij : α ijk : x l ÞÑ x l x l ÞÑ x l with 1 ď i j k ď n. Thus, IA 1 = t1u and IA 2 = Inn(F 2 ) F 2 are finitely presented. Krstić and McCool [1997]: IA 3 is not finitely presentable. It is not known whether IA n admits a finite presentation for n ě 4. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

24 AUTOMORPHISM GROUPS OF FREE GROUPS THE TORELLI GROUP OF F n Let I Fn = J 1 n = IA n be the Torelli group of F n. Recall we have an equivariant GL n (Z)-homomorphism, In degree 1, this can be written as J : gr F (IA n ) Ñ Der(L n ), J : gr 1 F (IA n) Ñ H b (H ^ H), where H = (F n ) ab = Z n, viewed as a GL n (Z)-module via the defining representation. Composing with ι F, we get a homomorphism J ι F : (IA n ) ab H b (H ^ H). THEOREM (ANDREADAKIS, COHEN PAKIANATHAN, FARB, KAWAZUMI) For each n ě 3, the map J ι F is a GL n (Z)-equivariant isomorphism. Thus, H 1 (IA n, Z) is free abelian, of rank b 1 (IA n ) = n 2 (n 1)/2. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

25 AUTOMORPHISM GROUPS OF FREE GROUPS We have a commuting diagram, Inn(F n ) = Inn(F n ) 1 IA n Aut(F n ) q q GL n (Z) 1 OA n Out(F n ) GL n (Z) 1 Thus, OA n = r I Fn. Write the induced Johnson filtration on Out(F n ) as r J s n = π(j s n). GL n (Z) acts on (OA n ) ab, and the outer Johnson homomorphism defines a GL n (Z)-equivariant isomorphism rj ι rf : (OA n ) ab H b (H ^ H)/H. = 1 Moreover, J r n 2 = OA 1 n, and we have an exact sequence 1 Fn 1 Ad IA 1 n OA 1 n 1. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

26 AUTOMORPHISM GROUPS OF FREE GROUPS DEEPER INTO THE JOHNSON FILTRATION CONJECTURE (F. COHEN, A. HEAP, A. PETTET 2010) If n ě 3, s ě 2, and 1 ď i ď n 2, the cohomology group H i (J s n, Z) is not finitely generated. We disprove this conjecture, at least rationally, in the case when n ě 5, s = 2, and i = 1. THEOREM If n ě 5, then dim Q H 1 (J 2 n, Q) ă 8. To start with, note that J 2 n = IA 1 n. Thus, it remains to prove that b 1 (IA 1 n) ă 8, i.e., (IA 1 n/ia 2 n) b Q is finite-dimensional. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

27 AUTOMORPHISM GROUPS OF FREE GROUPS REPRESENTATIONS OF sl n (C) h: the Cartan subalgebra of gl n (C), with coordinates t 1,..., t n. = tt i t i+1 1 ď i ď n 1u. λ i = t t i. V (λ): the irreducible, finite dimensional representation of sl n (C) with highest weight λ = ř iăn a iλ i, with a i P Z ě0. Set H C = H 1 (F n, C) = C n, and V := H 1 (OA n, C) = H C b (H C ^ H C )/H C. K K := ker ( Y: V ^ V Ñ H 2 (OA n, C) ). THEOREM (PETTET 2005) Let n ě 4. Set λ = λ 2 + λ n 1 (so that λ = λ 1 + λ n 2 ) and µ = λ 1 + λ n 2 + λ n 1. Then V = V (λ ) and K K = V (µ), as sl n (C)-modules. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

28 AUTOMORPHISM GROUPS OF FREE GROUPS THEOREM For each n ě 4, the resonance variety R(OA n ) vanishes. PROOF. 2λ µ = t 1 t n 1 is not a simple root. Thus, R(V, K ) = 0. REMARK When n = 3, the proof breaks down, since t 1 t 2 is a simple root. In fact, K K = V ^ V in this case. Thus, R(V, K ) = V. COROLLARY For each n ě 4, let V = V (λ 2 + λ n 1 ) and let K K = V (λ 1 + λ n 2 + λ n 1 ) Ă V ^ V be the Pettet summand. Then dim B(V, K ) ă 8 and dim gr q B(OA n ) ď dim B q (V, K ), for all q ě 0. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

29 AUTOMORPHISM GROUPS OF FREE GROUPS Using now a result of Dimca Papadima (2013) on the geometric irreducibility of representations of arithmetic groups, we obtain: THEOREM If n ě 4, then V(OA n ) is finite, and so b 1 (OA 1 n) ă 8. Finally, THEOREM If n ě 5, then b 1 (IA 1 n) ă 8. PROOF. The spectral sequence of the extension 1 gives rise to the exact sequence F 1 n IA 1 n OA 1 n H 1 (F 1 n, C) IA 1 n H 1 (IA 1 n, C) H 1 (OA 1 n, C) 0. The last term is finite-dimensional for all n ě 4, by previous theorem. The first term is finite-dimensional for all n ě 5, by nilpotency of the action of IA 1 n on F 1 n/f 2 n. Conclusion follows. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31 1

30 AUTOMORPHISM GROUPS OF FREE GROUPS TORELLI GROUPS OF SURFACES Let Σ g be a Riemann surface of genus g, and let I g = I π1 (Σ g ). I 1 is trivial, I 2 is not finitely generated. So assume g ě 3, in which case I g is finitely generated. Out + (π 1 (Σ g )) Ñ Sp 2g (Z) is surjective; thus, there is a natural Sp 2g (Z)-action on V = H 1 (I g, C). This action extends to a rational irrep of Sp 2g (C), and thus, of sp 2g (C). Dominant weights: λ i = t t i, for 1 ď i ď g. Let V = H 1 (I g, C), and let K K = ker(y) Ă V ^ V. Hain (1997): V = V (λ 3 ) and K K = V (2λ 2 ) V (0). Moreover, the decomposition of V ^ V into irreps is multiplicity-free. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31

31 AUTOMORPHISM GROUPS OF FREE GROUPS THEOREM R(I g ) = 0, for each g ě 4. PROOF. Simple roots: = tt 1 t 2, t 2 t 3,..., t g 1 t g, 2t g u. The only β P for which (λ 3, β) 0 is β = t 3 t 4. Clearly, 2λ 3 β = λ 2 + λ 4 is not a dominant weight for K K. Hence, R(V, K ) = 0. Let K g Ă I g be the Johnson kernel, i.e., the subgroup generated by Dehn twists about separating curves on Σ g. The above result (and some more work) implies the following: THEOREM (DIMCA PAPADIMA 2013) H 1 (K g, C) is finite-dimensional, for each g ě 4. ALEX SUCIU (NORTHEASTERN) RESONANCE, REPRESENTATIONS & JOHNSON U. CHICAGO, MAY / 31