Homological finiteness in the Andreadakis–Johnson filtration Alex Suciu Northeastern University Séminaire d’algèbre et de géométrie Université de Caen 21 juin, 2011 Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 1 / 26
Reference Stefan Papadima and Alexander I. Suciu, Homological finiteness in the Johnson filtration of the automorphism group of a free group , arxiv:1011.5292 Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 2 / 26
Outline The Johnson filtration 1 The Johnson homomorphism 2 The Torelli group of the free group 3 Alexander invariant and cohomology jump loci 4 Cohomology and sl n ( C ) -representation spaces 5 Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 3 / 26
The Johnson filtration Filtrations and graded Lie algebras Let G be a group, with commutator ( x , y ) = xyx − 1 y − 1 . Suppose given a descending filtration G = Φ 1 ⊇ Φ 2 ⊇ · · · ⊇ Φ s ⊇ · · · by subgroups of G , satisfying (Φ s , Φ t ) ⊆ Φ s + t , ∀ s , t ≥ 1 . Then Φ s ⊳ G , and Φ s / Φ s + 1 is abelian. Set � Φ s / Φ s + 1 . gr Φ ( G ) = s ≥ 1 This is a graded Lie algebra, with bracket [ , ]: gr s Φ × gr t Φ → gr s + t Φ induced by the group commutator. Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 4 / 26
The Johnson filtration Basic example: the lower central series , Γ s = Γ s ( G ) , defined as Γ 1 = G , Γ 2 = G ′ , . . . , Γ s + 1 = (Γ s , G ) , . . . Then for any filtration Φ as above, Γ s ⊆ Φ s ; thus, we have a morphism of graded Lie algebras, � gr Φ ( G ) . ι Φ : gr Γ ( G ) Example (P . Hall, E. Witt, W. Magnus) Let F n = � x 1 , . . . , x n � be the free group of rank n . Then: F n is residually nilpotent, i.e., � s ≥ 1 Γ s ( F n ) = { 1 } . gr Γ ( F n ) is isomorphic to the free Lie algebra L n = Lie ( Z n ) . � s gr s Γ ( F n ) is free abelian, of rank 1 d . d | s µ ( d ) n s If n ≥ 2, the center of L n is trivial. Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 5 / 26
The Johnson filtration Automorphism groups Let Aut ( G ) be the group of all automorphisms α : G → G , with α · β := α ◦ β . The Johnson filtration , Aut ( G ) = F 0 ⊇ F 1 ⊇ · · · ⊇ F s ⊇ · · · with terms F s = F s ( Aut ( G )) consisting of those automorphisms which act as the identity on the s -th nilpotent quotient of G : � Aut ( G ) → Aut ( G / Γ s + 1 � F s = ker = { α ∈ Aut ( G ) | α ( x ) · x − 1 ∈ Γ s + 1 , ∀ x ∈ G } ( F s , F t ) ⊆ F s + t . Kaloujnine [1950]: First term is the Torelli group , � � T G = F 1 = ker Aut ( G ) → Aut ( G ab ) . Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 6 / 26
The Johnson 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 ) → 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 { F s ( T G ) } s ≥ 1 , inherited from Aut ( G ) . The lower central series filtration, { Γ s ( T G ) } . The respective associated graded Lie algebras, gr F ( T G ) and gr Γ ( T G ) , come with natural actions of A ( G ) , and the morphism ι F : gr Γ ( T G ) → gr F ( T G ) is A ( G ) - equivariant. Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 7 / 26
The Johnson filtration Automorphism groups of free groups Identify ( F n ) ab = Z n , and Aut ( Z n ) = GL n ( Z ) . The homomorphism Aut ( F n ) → GL n ( Z ) is onto. Thus, A ( F n ) = GL n ( Z ) . Denote the Torelli group by IA n = T F n , 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 �→ x j x i x − 1 x i �→ x i · ( x j , x k ) j α ij : α ijk : x ℓ �→ x ℓ x ℓ �→ x ℓ with 1 ≤ i � = j � = k ≤ n . Thus, IA 1 = { 1 } and IA 2 = Inn ( F 2 ) ∼ = F 2 are finitely presented. Krsti´ c and McCool [1997]: IA 3 is not finitely presentable. It is not known whether IA n admits a finite presentation for n ≥ 4. Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 8 / 26
The Johnson filtration Nevertheless, IA n has some interesting finitely presented subgroups: The McCool group of “pure symmetric” automorphisms, P Σ n , generated by α ij , 1 ≤ i � = j ≤ n . The “upper triangular" McCool group, P Σ + n , generated by α ij , i > j . Cohen, Pakianathan, Vershinin, and Wu [2008]: P Σ + n = F n − 1 ⋊ · · · ⋊ F 2 ⋊ F 1 , with extensions by IA-automorphisms. The pure braid group, P n , consisting of those automorphisms in P Σ n that leave the word x 1 · · · x n ∈ F n invariant. P n = F n − 1 ⋊ · · · ⋊ F 2 ⋊ F 1 , with extensions by pure braid automorphisms. 2 ∼ = P 2 ∼ 3 ∼ = P 3 ∼ P Σ + P Σ + = Z , = F 2 × Z . Question (CPVW): Is P Σ + n ∼ = P n , for n ≥ 4? 4 �∼ Bardakov and Mikhailov [2008]: P Σ + = P 4 . Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 9 / 26
The Johnson homomorphism The Johnson homomorphism Given a graded Lie algebra g , let Der s ( g ) = { δ : g • → g • + s linear | δ [ x , y ] = [ δ x , y ] + [ x , δ y ] , ∀ x , y ∈ g } . Then Der ( g ) = � s ≥ 1 Der s ( g ) is a graded Lie algebra, with bracket [ δ, δ ′ ] = δ ◦ δ ′ − δ ′ ◦ δ . Theorem 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 ∈ F s ( T G ) and x ∈ Γ t ( G ) by α )(¯ x ) = α ( x ) · x − 1 . J (¯ Moreover, J is equivariant with respect to the natural actions of A ( G ) . Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 10 / 26
The Johnson homomorphism The Johnson homomorphism informs on the Johnson filtration. Theorem Let G be a group. For each q ≥ 1 , the following are equivalent: Γ ( T G ) → Der s ( gr Γ ( G )) is injective, for all s ≤ q. J ◦ ι F : gr s 1 Γ s ( T G ) = F s ( T G ) , for all s ≤ q + 1 . 2 Proposition Suppose G is residually nilpotent, gr Γ ( G ) is centerless, and Γ ( T G ) → Der 1 ( gr Γ ( G )) is injective. Then F 2 ( T G ) = T ′ J ◦ ι F : gr 1 G . Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 11 / 26
The Johnson homomorphism Let Inn ( G ) = im ( Ad : G → Aut ( G )) , where Ad x : G → G , y �→ xyx − 1 . Define the outer automorphism group of a group G by π � Inn ( G ) � Aut ( G ) � Out ( G ) � 1 . 1 Obtain: F s := π ( F s ) . Filtration { � � F s } s ≥ 0 on Out ( G ) : F 1 of Out ( G ) subgroup � T G = � The outer Torelli group of G : � � � Out ( G ) � A ( G ) � 1 . Exact sequence: 1 T G Let g be a graded Lie algebra, and ad : g → Der ( g ) , where ad x : g → g , y �→ [ x , y ] . Define the Lie algebra of outer derivations of g by q � im ( ad ) � Der ( g ) � � � 0 . 0 Der ( g ) Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 12 / 26
�� � �� � �� � � � � � The Johnson homomorphism Theorem Suppose Z ( gr Γ ( G )) = 0 . Then the Johnson homomorphism induces an A ( G ) -equivariant monomorphism of graded Lie algebras, � F ( � � � J : gr � T G ) Der ( gr Γ ( G )) . To summarize: = = gr Γ ( G ) gr Γ ( G ) gr Γ ( G ) � � � � � � gr Γ ( Ad ) ad Ad ι F gr F ( T G ) � � J gr Γ ( T G ) Der ( gr Γ ( G )) gr Γ ( π ) q π ¯ ι � � J gr Γ ( � � gr � F ( � T G ) � � � � F T G ) Der ( gr Γ ( G )) , Alex Suciu (Northeastern University) The Andreadakis–Johnson filtration U. Caen, June 2011 13 / 26
The Torelli group of the free group The Torelli group of F n Let T F n = J 1 n = IA n be the Torelli group of F n . Recall we have an equivariant GL n ( Z ) -homomorphism, J : gr F ( IA n ) → Der ( L n ) , In degree 1, this can be written as F ( IA n ) → H ∗ ⊗ ( H ∧ H ) , J : gr 1 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 � H ∗ ⊗ ( H ∧ H ) . J ◦ ι F : ( IA n ) ab Theorem (Andreadakis, Cohen–Pakianathan, Farb, Kawazumi) For each n ≥ 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 University) The Andreadakis–Johnson filtration U. Caen, June 2011 14 / 26
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