Automorphism groups, Lie algebras, and resonance varieties

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

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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



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O UTLINE

1

T HE J OHNSON FILTRATION

2

A LEXANDER INVARIANTS

3

R ESONANCE VARIETIES

4

R OOTS ,

5

A UTOMORPHISM GROUPS OF FREE GROUPS

WEIGHTS , AND VANISHING RESONANCE



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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 = Φ1 Ě Φ2 Ě ¨ ¨ ¨ Ě Φs Ě ¨ ¨ ¨ by subgroups of G, satisfying

( Φ s , Φ t ) Ď Φ s +t ,

@s, t ě 1.

Then Φs Ÿ G, and Φs /Φs +1 is abelian. Set à s grΦ (G ) = Φ /Φs +1 . sě1

This is a graded Lie algebra, with bracket [ , ] : grsΦ ˆ grtΦ Ñ grsΦ+t induced by the group commutator. A LEX S UCIU (N ORTHEASTERN )


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Basic example: the lower central series, Γs = Γs (G ), defined as Γ1 = G, Γ2 = G1 , . . . , Γs +1 = (Γs , G ), . . . Then for any filtration Φ as above, Γs Ď Φs ; thus, we have a morphism of graded Lie algebras, ι Φ : grΓ (G )

/ gr (G ) . Φ

E XAMPLE (P. H ALL , E. W ITT, W. M AGNUS ) Let Fn = xx1 , . . . , xn y be the free group of rank n. Then: Ş Fn is residually nilpotent, i.e., sě1 Γs (Fn ) = t1u. grΓ (Fn ) is isomorphic to the free Lie algebra Ln = Lie(Zn ). ř s grsΓ (Fn ) is free abelian, of rank 1s d|s µ(d )n d . If n ě 2, the center of Ln is trivial. A LEX S UCIU (N ORTHEASTERN )


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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/Γs +1

= tα P Aut(G ) | α(x ) ¨ x ´1 P Γs+1 , @x P Gu Kaloujnine [1950]: (F s , F t ) Ď F s +t . First term is the Torelli group, TG = F 1 = ker Aut(G ) Ñ Aut(Gab ) . A LEX S UCIU (N ORTHEASTERN )


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By construction, F 1 = TG is a normal subgroup of F 0 = Aut(G ). The quotient group,

A(G ) = F 0 /F 1 = im(Aut(G ) Ñ Aut(Gab )) is the symmetry group of TG ; it fits into exact sequence 1

/ TG

/ Aut(G )

/ A(G )

/1.

The Torelli group comes endowed with two filtrations: The Johnson filtration tF s (TG )usě1 , inherited from Aut(G ). The lower central series filtration, tΓs (TG )u. The respective associated graded Lie algebras, grF (TG ) and grΓ (TG ), come endowed with natural actions of A(G ); moreover, the morphism ιF : grΓ (TG ) Ñ grF (TG ) is A(G )-equivariant.



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T HE J OHNSON HOMOMORPHISM Given a graded Lie algebra g, let Ders (g) = tδ : g‚ Ñ g‚+s linear | δ[x, y ] = [δx, y ] + [x, δy ], @x, y P gu. À Then Der(g) = sě1 Ders (g) is a graded Lie algebra, with bracket [δ, δ1 ] = δ ˝ δ1 ´ δ1 ˝ δ. T HEOREM Given a group G, there is a monomorphism of graded Lie algebras, J : grF (TG )

/ Der(gr (G )) , Γ

given on homogeneous elements α P F s (TG ) and x P Γt (G ) by J (α¯ )(x¯ ) = α(x ) ¨ x ´1 . Moreover, J is equivariant with respect to the natural actions of A(G ). A LEX S UCIU (N ORTHEASTERN )


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The Johnson homomorphism informs on the Johnson filtration. T HEOREM Let G be a group. For each q ě 1, the following are equivalent: 1

J ˝ ιF : grsΓ (TG ) Ñ Ders (grΓ (G )) is injective, for all s ď q.

2

Γs (TG ) = F s (TG ), for all s ď q + 1.

P ROPOSITION Suppose G is residually nilpotent, grΓ (G ) is centerless, and J ˝ ιF : gr1Γ (TG ) Ñ Der1 (grΓ (G )) is injective. Then F 2 (TG ) = TG1 . P ROBLEM Determine the homological finiteness properties of the groups F s (TG ). In particular, decide whether dim H1 (TG1 , Q) ă 8. A LEX S UCIU (N ORTHEASTERN )


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A N OUTER VERSION Let Inn(G ) = im(Ad : G Ñ Aut(G )), where Adx : G Ñ G, y ÞÑ xyx ´1 . Define the outer automorphism group of a group G by 1

/ Aut(G )

/ Inn(G )

π

/ Out(G )

/1.

We then have r s usě0 on Out(G ): F r s : = π (F s ). Filtration tF r 1 of Out(G ). The outer Torelli group of G: subgroup TrG = F / Out(G ) / TrG / A(G ) /1. Exact sequence: 1 T HEOREM Suppose Z (grΓ (G )) = 0. Then the Johnson homomorphism induces an A(G )-equivariant monomorphism of graded Lie algebras, Ą (gr (G )) , / Der Jr : gr r (TrG ) F

Γ

Ą (g) = Der(g)/ im(ad : g Ñ Der(g)). where Der A LEX S UCIU (N ORTHEASTERN )


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T HE A LEXANDER INVARIANT Let G be a group, and Gab = G/G1 its maximal abelian quotient. Let G2 = (G1 , G1 ); then G/G2 is the maximal metabelian quotient. / G1 /G2 / G/G2 / Gab /0. Get exact sequence 0 Conjugation in G/G2 turns the abelian group B (G ) := G1 /G2 = H1 (G1 , Z) into a module over R = ZGab , called the Alexander invariant of G. Since both G1 and G2 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 TG acts R-linearly on the Alexander invariant B (G ). A LEX S UCIU (N ORTHEASTERN )


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C HARACTERISTIC VARIETIES Let G be a finitely generated group. p = Hom(G, C˚ ) be its character group: an algebraic group, Let G with coordinate ring C[Gab ]. p ab Ý» p The map ab : G Gab induces an isomorphism G Ñ G. ˝ ˚ n p – (C ) , where n = rank Gab . G D EFINITION The (first) characteristic variety of G is the support of the (complexified) Alexander invariant B = B (G ) b C: p V (G ) := V (ann B ) Ă G. This variety informs on the Betti numbers of normal subgroups H Ÿ G with G/H abelian. In particular (for H = G1 ): P ROPOSITION The set V (G ) is finite if and only if b1 (G1 ) = dimC B (G ) b C is finite. A LEX S UCIU (N ORTHEASTERN )


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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.



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K OSZUL MODULES Let S = Sym Ź (V ) be the symmetric algebra on V . Let (S bC V , δ) be the Koszul resolution, with differential Ź Ź δp : S bC p V Ñ S bC p´1 V given by ÿp (´1)j´1 vij b (vi1 ^ ¨ ¨ ¨ ^ vpij ^ ¨ ¨ ¨ ^ vip ). vi1 ^ ¨ ¨ ¨ ^ vip ÞÑ j =1

Let ι : K Ñ V ^ V be the inclusion map. The Koszul module B(V , K ) is the graded S-module presented as δ3 +id bι Ź3 Ź / S bC 2 V / / B(V , K ) . S bC V ‘K 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 dimC B ă 8. A LEX S UCIU (N ORTHEASTERN )


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C OHOMOLOGY JUMP LOCI Let A = A(V , K ) be the quadratic algebra defined as the quotient Ź of the exterior algebra E = V ˚ by the ideal generated by K K Ă V ˚ ^ V ˚ = E 2. Then R is the set of points a P A1 where the cochain complex A0

a

/ A1

a

/ A2

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:

Bq˚ – TorEq+1 (A, C)q +2



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VANISHING RESONANCE Setting m = dim K , we may view K as a point in the Grassmannian Grm (V ^ V ), and P(K K ) as a codimension m projective subspace in P(V ˚ ^ V ˚ ). L EMMA Let Gr2 (V ˚ ) ãÑ P(V ˚ ^ V ˚ ) be the Plücker embedding. Then,

R(V , K ) = 0 ðñ P(K K ) X Gr2 (V ˚ ) = H. T HEOREM For any integer m with 0 ď m ď (n2), where n = dim V , the set ( Un,m = K P Grm (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 Bq (V , K ) = 0, for every K P Un,m . A LEX S UCIU (N ORTHEASTERN )


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R ESONANCE VARIETIES OF GROUPS The resonance variety of a f.g. group G is denied as R(G ) = R(V , K ), where V ˚ = H 1 (G, C) and K K = ker(YG : V ˚ ^ V ˚ Ñ H 2 (G, C)). Rationally, every resonance variety arises in this fashion: P ROPOSITION 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 G with V ˚ = H 1 (G, C) and K K = ker(YG ). The resonance variety R = R(G ) can be viewed as an approximation to the characteristic variety V = V (G ). T HEOREM (L IBGOBER , D IMCA –PAPADIMA –S.) Let TC1 (V ) be the tangent cone to V at 1, viewed as a subset of p ) = H 1 (G, C). Then TC1 (V ) Ď R. Moreover, if G is 1-formal, T1 (G then equality holds, and R is a union of rational subspaces. A LEX S UCIU (N ORTHEASTERN )


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E XAMPLE (R IGHT- ANGLED A RTIN GROUPS ) Let Γ = (V, E) be a (finite, simple) graph. The corresponding right-angled Artin group is GΓ = xv P V | vw = wv if tv , wu P Ey. V = H1 (GΓ , 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(GΓ ) Ă CV is the union of all coordinate subspaces CW corresponding to subsets W Ă V for which the induced graph ΓW is disconnected. ř The Hilbert series qě0 dimC (Bq )t q +2 equals QΓ (t /(1 ´ t )), where QΓř(t ) is the “cut polynomial" of Γ, with coefficient of t k equal to WĂV : |W|=k b˜ 0 (ΓW ), where b˜ 0 (ΓW ) is one less than the number of components of the induced subgraph on W. A LEX S UCIU (N ORTHEASTERN )


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R OOTS , WEIGHTS , AND VANISHING RESONANCE

R OOTS , 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 xβ , yβ P g and hβ P h which generate a subalgebra of g isomorphic to sl2 (C). Each irreducible representation of g is of the form V (λ), where λ is a dominant weight. A non-zero vector v P V (λ) is a maximal vector (of weight λ) if xβ ¨ v = 0, for all β P ∆. Such a vector is uniquely determined (up to non-zero scalars), and is denoted by vλ . L EMMA 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. A LEX S UCIU (N ORTHEASTERN )


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T HEOREM 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 vλ˚ be a maximal vector for V ˚ . 1

2

Suppose there is a root β P ∆ such that (λ˚ , β) ‰ 0, and suppose the vector vλ˚ ^ yβ vλ˚ (of weight 2λ˚ ´ β) belongs to K K . Then R(V , K ) ‰ 0. Suppose that 2λ˚ ´ β is not a dominant weight for K K , for any simple root β. Then R(V , K ) = 0.

C OROLLARY

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.



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T HE CASE OF g = sl2 (C) h˚ is spanned t1 and t2 (the dual coordinates on the subspace of diagonal 2 ˆ 2 complex matrices), subject to t1 + t2 = 0. There is a single simple root, β = t1 ´ t2 . The defining representation is V (λ1 ), where λ1 = t1 . The irreps are of the form Vn = V (nλ1 ) = Symn (V (λ1 )), for some n ě 0. Moreover, dim Vn = n + 1 and Vn˚ = Vn . The second exterior power of Vn decomposes into irreducibles, according to the Clebsch-Gordan rule: à Vn ^ Vn = V2n´2´4j . jě0

These summands occur with multiplicity 1, and V2n´2 is always one of those summands. A LEX S UCIU (N ORTHEASTERN )


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P ROPOSITION Let K be an sl2 (C)-submodule of Vn ^ Vn . TFAE: 1

The variety R(Vn , K ) consists only of 0 P Vn˚ .

2

The C-vector space B(Vn , K ) is finite-dimensional.

3

The representation K contains V2n´2 as a direct summand.

The Sym(Vn )-modules W (n) := B(Vn , V2n´2 ) have been studied by J. Weyman and D. Eisenbud (1990). We recover and strengthen one of their results: C OROLLARY For any sl2 (C)-submodule K Ă Vn ^ Vn , the Koszul module B(Vn , K ) is finite-dimensional over C if and only if B(Vn , K ) is a quotient of W (n). The vanishing of Wn´2 (n), for all n ě 1, implies the generic Green Conjecture on free resolutions of canonical curves. The determination of the Hilbert series of the Weyman modules W (n) remains an interesting open problem. A LEX S UCIU (N ORTHEASTERN )


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A UTOMORPHISM GROUPS OF FREE GROUPS Identify (Fn )ab = Zn , and Aut(Zn ) = GLn (Z). The morphism Aut(Fn ) Ñ GLn (Z) is onto; thus, A(Fn ) = GLn (Z). Denote the Torelli group by IAn = TFn , and the Johnson–Andreadakis filtration by Jns = F s (Aut(Fn )). Magnus [1934]: IAn is generated by the automorphisms # # xi ÞÑ xj xi xj´1 xi ÞÑ xi ¨ (xj , xk ) αij : αijk : x` ÞÑ x` x` ÞÑ x` with 1 ď i ‰ j ‰ k ď n. Thus, IA1 = t1u and IA2 = Inn(F2 ) – F2 are finitely presented. Krsti´c and McCool [1997]: IA3 is not finitely presentable. It is not known whether IAn admits a finite presentation for n ě 4. A LEX S UCIU (N ORTHEASTERN )


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Nevertheless, IAn has some interesting finitely presented subgroups: The McCool group of “pure symmetric” automorphisms, PΣn , generated by αij , 1 ď i ‰ j ď n. The “upper triangular" McCool group, PΣn+ , generated by αij , i ą j. Cohen, Pakianathan, Vershinin, and Wu [2008]: PΣn+ = Fn´1 ¸ ¨ ¨ ¨ ¸ F2 ¸ F1 , with extensions by IA-automorphisms. The pure braid group, Pn , consisting of those automorphisms in PΣn that leave the word x1 ¨ ¨ ¨ xn P Fn invariant. Pn = Fn´1 ¸ ¨ ¨ ¨ ¸ F2 ¸ F1 , with extensions by pure braid automorphisms. PΣ2+ – P2 – Z,

PΣ3+ – P3 – F2 ˆ Z.

Question (CPVW): Is PΣn+ – Pn , for n ě 4? Bardakov and Mikhailov [2008]: PΣ4+ fl P4 . A LEX S UCIU (N ORTHEASTERN )


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T HE T ORELLI GROUP OF Fn Let TFn = Jn1 = IAn be the Torelli group of Fn . Recall we have an equivariant GLn (Z)-homomorphism, J : grF (IAn ) Ñ Der(Ln ), In degree 1, this can be written as J : gr1F (IAn ) Ñ H ˚ b (H ^ H ), where H = (Fn )ab = Zn , viewed as a GLn (Z)-module via the defining representation. Composing with ιF , we get a homomorphism J ˝ ιF : (IAn )ab

/ H ˚ b (H ^ H ) .

T HEOREM (A NDREADAKIS , C OHEN –PAKIANATHAN , FARB , K AWAZUMI ) For each n ě 3, the map J ˝ ιF is a GLn (Z)-equivariant isomorphism. Thus, H1 (IAn , Z) is free abelian, of rank b1 (IAn ) = n2 (n ´ 1)/2. A LEX S UCIU (N ORTHEASTERN )


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We have a commuting diagram, _

/ Inn(Fn ) _

/ IAn

/ Aut(Fn )

Inn(F n) 1 1

π

/ OAn

=

π

/ Out(Fn )

/ GLn (Z)

/1

=

/ GLn (Z)

/1

Thus, OAn = TrFn . Write the induced Johnson filtration on Out(Fn ) as Jrns = π (Jns ). GLn (Z) acts on (OAn )ab , and the outer Johnson homomorphism defines a GLn (Z)-equivariant isomorphism Jr ˝ ιFr : (OAn )ab

–

/ H ˚ b (H ^ H )/H .

Moreover, Jrn2 = OA1n , and we have an exact sequence 1 A LEX S UCIU (N ORTHEASTERN )

/ F 1 Ad / IA1 n n

/ OA1 n


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D EEPER INTO THE J OHNSON FILTRATION C ONJECTURE (F. C OHEN , A. H EAP, A. P ETTET 2010) If n ě 3, s ě 2, and 1 ď i ď n ´ 2, the cohomology group H i (Jns , Z) is not finitely generated. We disprove this conjecture, at least rationally, in the case when n ě 5, s = 2, and i = 1. T HEOREM If n ě 5, then dimQ H 1 (Jn2 , Q) ă 8. To start with, note that Jn2 = IA1n . Thus, it remains to prove that b1 (IA1n ) ă 8, i.e., (IA1n /IA2n ) b Q is finite dimensional. A LEX S UCIU (N ORTHEASTERN )


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R EPRESENTATIONS OF sln (C) h: the Cartan subalgebra of gln (C), with coordinates t1 , . . . , tn . ∆ = tti ´ ti +1 | 1 ď i ď n ´ 1u. λ i = t1 + ¨ ¨ ¨ + ti . V (λ): the irreducible, finite ř dimensional representation of sln (C) with highest weight λ = iăn ai λi , with ai P Zě0 . Set HC = H1 (Fn , C) = Cn , and V ˚ := H 1 (OAn , C) = HC b (HC˚ ^ HC˚ )/HC˚ . K K := ker Y : V ˚ ^ V ˚ Ñ H 2 (OAn , C) . T HEOREM (P ETTET 2005) Let n ě 4. Set λ = λ2 + λn´1 (so that λ˚ = λ1 + λn´2 ) and µ = λ1 + λn´2 + λn´1 . Then V ˚ = V (λ˚ ) and K K = V (µ), as sln (C)-modules. A LEX S UCIU (N ORTHEASTERN )


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T HEOREM For each n ě 4, the resonance variety R(OAn ) vanishes. P ROOF . 2λ˚ ´ µ = t1 ´ tn´1 is not a simple root. Thus, R(V , K ) = 0. R EMARK When n = 3, the proof breaks down, since t1 ´ t2 is a simple root. In fact, K K = V ˚ ^ V ˚ in this case, and so R(V , K ) = V ˚ . C OROLLARY For each n ě 4, let V = V (λ2 + λn´1 ) and let K K = V (λ1 + λn´2 + λn´1 ) Ă V ˚ ^ V ˚ be the Pettet summand. Then dim B(V , K ) ă 8 and dim grq B (OAn ) ď dim Bq (V , K ), for all q ě 0.



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Using now a result of Dimca–Papadima on the “geometric irreducibility” of representations of arithmetic groups, we obtain: T HEOREM If n ě 4, then V (OAn ) is finite, and so b1 (OA1n ) ă 8. Finally, T HEOREM If n ě 5, then b1 (IA1n ) ă 8. P ROOF . The spectral sequence of the extension 1 gives rise to the exact sequence H1 (Fn1 , C)IA1n

/ H1 (IA1 , C) n

/ F1 n

/ IA1 n

/ H1 (OA1 , C) n

/ OA1

n

/1

/0.

The last term is finite-dimensional for all n ě 4 by the previous theorem, while the first term is finite-dimensional for all n ě 5, by the nilpotency of the action of IA1n on Fn1 /Fn2 . A LEX S UCIU (N ORTHEASTERN )


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