The Euler characteristic of Out( F n ) and the Hopf algebra of - PowerPoint PPT Presentation

Further motivation to look at Euler characteristic of Out( F n ) Consider the abelization map F n → Z n . ⇒ Induces a group homomorphism 1 → T n → Out( F n ) → Out( Z n ) → 1 � �� � =GL( n , Z ) • T n the ‘non-abelian’ part of Out( F n ) is interesting. • By the short exact sequence above n ≥ 3 χ (Out( F n )) = χ (GL( n , Z )) χ ( T n ) � �� � =0 ⇒ T n does not have finitely-generated homology for n ≥ 3 if χ (Out( F n )) � = 0. 14

Conjectures Conjecture Smillie-Vogtmann (1987) χ (Out( F n )) � = 0 for all n ≥ 2 and | χ (Out( F n )) | grows exponentially for n → ∞ . based on initial computations by Smillie-Vogtmann (1987) up to n ≤ 11. Later strengthened by Zagier (1989) up to n ≤ 100. 15

Conjectures Conjecture Smillie-Vogtmann (1987) χ (Out( F n )) � = 0 for all n ≥ 2 and | χ (Out( F n )) | grows exponentially for n → ∞ . based on initial computations by Smillie-Vogtmann (1987) up to n ≤ 11. Later strengthened by Zagier (1989) up to n ≤ 100. Conjecture Magnus (1934) T n is not finitely presentable. 15

Conjectures Conjecture Smillie-Vogtmann (1987) χ (Out( F n )) � = 0 for all n ≥ 2 and | χ (Out( F n )) | grows exponentially for n → ∞ . based on initial computations by Smillie-Vogtmann (1987) up to n ≤ 11. Later strengthened by Zagier (1989) up to n ≤ 100. Conjecture Magnus (1934) T n is not finitely presentable. In topological terms, i.e. dim( H 2 ( T n )) = ∞ , which implies that T n does not have finitely-generated homology. 15

Conjectures Conjecture Smillie-Vogtmann (1987) χ (Out( F n )) � = 0 for all n ≥ 2 and | χ (Out( F n )) | grows exponentially for n → ∞ . based on initial computations by Smillie-Vogtmann (1987) up to n ≤ 11. Later strengthened by Zagier (1989) up to n ≤ 100. Conjecture Magnus (1934) T n is not finitely presentable. In topological terms, i.e. dim( H 2 ( T n )) = ∞ , which implies that T n does not have finitely-generated homology. Theorem Bestvina, Bux, Margalit (2007) T n does not have finitely-generated homology. 15

Results: χ (Out( F n )) � = 0

Theorem A MB-Vogtmann (2019) χ (Out( F n )) < 0 for all n ≥ 2 16

Theorem A MB-Vogtmann (2019) χ (Out( F n )) < 0 for all n ≥ 2 1 Γ ( n − 3 / 2) √ χ (Out( F n )) ∼ − as n → ∞ . log 2 n 2 π 16

Theorem A MB-Vogtmann (2019) χ (Out( F n )) < 0 for all n ≥ 2 1 Γ ( n − 3 / 2) √ χ (Out( F n )) ∼ − as n → ∞ . log 2 n 2 π which settles the initial conjecture by Smillie-Vogtmann (1987) . Immediate questions: 16

Theorem A MB-Vogtmann (2019) χ (Out( F n )) < 0 for all n ≥ 2 1 Γ ( n − 3 / 2) √ χ (Out( F n )) ∼ − as n → ∞ . log 2 n 2 π which settles the initial conjecture by Smillie-Vogtmann (1987) . Immediate questions: ⇒ Huge amount of unstable homology in odd dimensions. 16

Theorem A MB-Vogtmann (2019) χ (Out( F n )) < 0 for all n ≥ 2 1 Γ ( n − 3 / 2) √ χ (Out( F n )) ∼ − as n → ∞ . log 2 n 2 π which settles the initial conjecture by Smillie-Vogtmann (1987) . Immediate questions: ⇒ Huge amount of unstable homology in odd dimensions. • Only one odd-dimensional class known Bartholdi (2016) . 16

Theorem A MB-Vogtmann (2019) χ (Out( F n )) < 0 for all n ≥ 2 1 Γ ( n − 3 / 2) √ χ (Out( F n )) ∼ − as n → ∞ . log 2 n 2 π which settles the initial conjecture by Smillie-Vogtmann (1987) . Immediate questions: ⇒ Huge amount of unstable homology in odd dimensions. • Only one odd-dimensional class known Bartholdi (2016) . • Where does all this homology come from? 16

This Theorem A follows from an implicit expression for χ (Out( F n )): 17

This Theorem A follows from an implicit expression for χ (Out( F n )): Theorem B MB-Vogtmann (2019) √ � 2 π e − N N N ∼ a k ( − 1) k Γ ( N + 1 / 2 − k ) as N → ∞ k ≥ 0   � � a k z k = exp χ (Out( F n +1 )) z n  where n ≥ 0 k ≥ 0 17

This Theorem A follows from an implicit expression for χ (Out( F n )): Theorem B MB-Vogtmann (2019) √ � 2 π e − N N N ∼ a k ( − 1) k Γ ( N + 1 / 2 − k ) as N → ∞ k ≥ 0   � � a k z k = exp χ (Out( F n +1 )) z n  where n ≥ 0 k ≥ 0 ⇒ χ (Out( F n )) are the coe ffi cients of an asymptotic expansion. 17

This Theorem A follows from an implicit expression for χ (Out( F n )): Theorem B MB-Vogtmann (2019) √ � 2 π e − N N N ∼ a k ( − 1) k Γ ( N + 1 / 2 − k ) as N → ∞ k ≥ 0   � � a k z k = exp χ (Out( F n +1 )) z n  where n ≥ 0 k ≥ 0 ⇒ χ (Out( F n )) are the coe ffi cients of an asymptotic expansion. • An analytic argument is needed to prove Theorem A from Theorem B. 17

This Theorem A follows from an implicit expression for χ (Out( F n )): Theorem B MB-Vogtmann (2019) √ � 2 π e − N N N ∼ a k ( − 1) k Γ ( N + 1 / 2 − k ) as N → ∞ k ≥ 0   � � a k z k = exp χ (Out( F n +1 )) z n  where n ≥ 0 k ≥ 0 ⇒ χ (Out( F n )) are the coe ffi cients of an asymptotic expansion. • An analytic argument is needed to prove Theorem A from Theorem B. • In this talk: Focus on proof of Theorem B 17

Analogy to the mapping class group

Harer-Zagier formula for χ (MCG( S g )) Similar result for the mapping class group/moduli space of curves: 18

Harer-Zagier formula for χ (MCG( S g )) Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) B 2 g g ≥ 2 χ ( M g ) = χ (MCG( S g )) = 4 g ( g − 1) 18

Harer-Zagier formula for χ (MCG( S g )) Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) B 2 g g ≥ 2 χ ( M g ) = χ (MCG( S g )) = 4 g ( g − 1) • Original proof by Harer and Zagier in 1986. 18

Harer-Zagier formula for χ (MCG( S g )) Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) B 2 g g ≥ 2 χ ( M g ) = χ (MCG( S g )) = 4 g ( g − 1) • Original proof by Harer and Zagier in 1986. • Alternative proof using topological field theory (TFT) by Penner (1988) . 18

Harer-Zagier formula for χ (MCG( S g )) Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) B 2 g g ≥ 2 χ ( M g ) = χ (MCG( S g )) = 4 g ( g − 1) • Original proof by Harer and Zagier in 1986. • Alternative proof using topological field theory (TFT) by Penner (1988) . • Simplified proof by Kontsevich (1992) based on TFT’s. 18

Harer-Zagier formula for χ (MCG( S g )) Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) B 2 g g ≥ 2 χ ( M g ) = χ (MCG( S g )) = 4 g ( g − 1) • Original proof by Harer and Zagier in 1986. • Alternative proof using topological field theory (TFT) by Penner (1988) . • Simplified proof by Kontsevich (1992) based on TFT’s. ⇒ Kontsevich’s proof served as a blueprint for χ (Out( F n )). 18

Kontsevich’s argument 19

Kontsevich’s argument • We have the identity by Kontsevich (1992) : � � ( − 1) | V G | χ ( M g , n ) z 2 − 2 g − n = | Aut G | z χ ( G ) . n ! g , n connected graphs G 19

Kontsevich’s argument • We have the identity by Kontsevich (1992) : � � ( − 1) | V G | χ ( M g , n ) z 2 − 2 g − n = | Aut G | z χ ( G ) . n ! g , n connected graphs G • Kontsevich proved this using a combinatorial model of M g , n by Penner (1986) based on ribbon graphs. 19

Kontsevich’s argument • We have the identity by Kontsevich (1992) : � � ( − 1) | V G | χ ( M g , n ) z 2 − 2 g − n = | Aut G | z χ ( G ) . n ! g , n connected graphs G • Kontsevich proved this using a combinatorial model of M g , n by Penner (1986) based on ribbon graphs. 1xlr1 I h.COM 1 get n M 19

Kontsevich’s argument • We have the identity by Kontsevich (1992) : � � ( − 1) | V G | χ ( M g , n ) z 2 − 2 g − n = | Aut G | z χ ( G ) . n ! g , n connected graphs G • Kontsevich proved this using a combinatorial model of M g , n by Penner (1986) based on ribbon graphs. • The expression on the right hand side can be evaluated using a ‘topological field theory’: � � � � ( − 1) | V G | 1 | Aut G | z χ ( G ) = log e z (1+ x − e x ) dx √ 2 π z R connected graphs G � ζ ( − k ) z − k = − k k ≥ 1 19

Kontsevich’s argument • We have the identity by Kontsevich (1992) : � � ( − 1) | V G | χ ( M g , n ) z 2 − 2 g − n = | Aut G | z χ ( G ) . n ! g , n connected graphs G • Kontsevich proved this using a combinatorial model of M g , n by Penner (1986) based on ribbon graphs. • The expression on the right hand side can be evaluated using a ‘topological field theory’: � � � � ( − 1) | V G | 1 | Aut G | z χ ( G ) = log e z (1+ x − e x ) dx √ 2 π z R connected graphs G I � ζ ( − k ) z − k = TFT − k k ≥ 1 action 19

Kontsevich’s argument • We have the identity by Kontsevich (1992) : � � ( − 1) | V G | χ ( M g , n ) z 2 − 2 g − n = | Aut G | z χ ( G ) . n ! g , n connected graphs G • Kontsevich proved this using a combinatorial model of M g , n by Penner (1986) based on ribbon graphs. • The expression on the right hand side can be evaluated using a ‘topological field theory’: � � � � ( − 1) | V G | 1 | Aut G | z χ ( G ) = log e z (1+ x − e x ) dx √ 2 π z R connected graphs G � ζ ( − k ) z − k = − k k ≥ 1 • The formula for χ ( M g , n ) follows via the short exact sequence 1 → π 1 ( S g , n ) → M g , n +1 → M g , n → 1 19

Kontsevich’s argument • We have the identity by Kontsevich (1992) : � � ( − 1) | V G | χ ( M g , n ) z 2 − 2 g − n = | Aut G | z χ ( G ) . n ! g , n connected graphs G • Kontsevich proved this using a combinatorial model of M g , n by Penner (1986) based on ribbon graphs. • The expression on the right hand side can be evaluated using a ‘topological field theory’: � � � � ( − 1) | V G | 1 | Aut G | z χ ( G ) = log e z (1+ x − e x ) dx √ 2 π z R connected graphs G � ζ ( − k ) z − k = − k k ≥ 1 • The formula for χ ( M g , n ) follows via the short exact sequence 1 → π 1 ( S g , n ) → M g , n +1 → M g , n → 1 19

An algebraic viewpoint 20

An algebraic viewpoint • Let H be the Q -vector space spanned by a set of graphs: ∅ H = Q + Q + Q + . . . ���� =: I 20

An algebraic viewpoint • Let H be the Q -vector space spanned by a set of graphs: ∅ H = Q + Q + Q + . . . ���� =: I • Here: only connected graphs with 3-or-higher-valent vertices. 20

An algebraic viewpoint • Let H be the Q -vector space spanned by a set of graphs: ∅ H = Q + Q + Q + . . . ���� =: I • Here: only connected graphs with 3-or-higher-valent vertices. � χ ( M g , n ) z 2 − 2 g − n = φ ( X ) n ! g , n where � G | Aut G | z χ ( G ) ∈ H [[ z − 1 ]] X := G and φ : H → Q , G → ( − 1) | V G | 20

An algebraic viewpoint • Let H be the Q -vector space spanned by a set of graphs: ∅ H = Q + Q + Q + . . . ���� =: I • Here: only connected graphs with 3-or-higher-valent vertices. � χ ( M g , n ) z 2 − 2 g − n = φ ( X ) n ! g , n where � G | Aut G | z χ ( G ) ∈ H [[ z − 1 ]] X := G and φ : H → Q , G → ( − 1) | V G | ⇒ φ is very simple and easy to handle via topological field theory. 20

• For Out( F n ), we find that � χ (Out( F n +1 )) z − n = τ ( X ) n ≥ 1 with X as before and � ( − 1) | E f | τ : H → Q , G → f ⊂ G where the sum is over all forests (acyclic subgraphs) of G . 21

• For Out( F n ), we find that � χ (Out( F n +1 )) z − n = τ ( X ) n ≥ 1 with X as before and � ( − 1) | E f | τ : H → Q , G → f ⊂ G where the sum is over all forests (acyclic subgraphs) of G . ⇒ Not directly approachable with a TFT... 21

• For Out( F n ), we find that � χ (Out( F n +1 )) z − n = τ ( X ) n ≥ 1 with X as before and � ( − 1) | E f | τ : H → Q , G → f ⊂ G where the sum is over all forests (acyclic subgraphs) of G . ⇒ Not directly approachable with a TFT... • The necessary combinatorial model is the ‘forest collapse’ construction by Culler-Vogtmann (1986) . 21

The Hopf algebra of graphs 22

The Hopf algebra of graphs • With disjoint union of graphs m : H ⊗ H → H , G 1 ⊗ G 2 �→ G 1 ⊎ G 2 as multiplication, the empty graph ∅ associated with the neutral element I , 22

The Hopf algebra of graphs • With disjoint union of graphs m : H ⊗ H → H , G 1 ⊗ G 2 �→ G 1 ⊎ G 2 as multiplication, the empty graph ∅ associated with the neutral element I , • and the coproduct ∆ : H → H ⊗ H , � ∆ : G �→ g ⊗ G / g , g ⊂ G bridgeless g where the sum is over all bridgeless subgraphs, 22

The Hopf algebra of graphs • With disjoint union of graphs m : H ⊗ H → H , G 1 ⊗ G 2 �→ G 1 ⊎ G 2 as multiplication, the empty graph ∅ associated with the neutral element I , • and the coproduct ∆ : H → H ⊗ H , � ∆ : G �→ g ⊗ G / g , g ⊂ G bridgeless g where the sum is over all bridgeless subgraphs, • the vector space H becomes the core Hopf algebra of graphs Kreimer (2009) , which is closely related to the Hopf algebra of renormalization in quantum field theory. 22

23

• Characters, i.e. linear maps ψ : H → A which fulfill ψ ( I ) = I A form a group under the convolution product, ψ ⋆ µ = m ◦ ( ψ ⊗ µ ) ◦ ∆ 23

• Characters, i.e. linear maps ψ : H → A which fulfill ψ ( I ) = I A form a group under the convolution product, ψ ⋆ µ = m ◦ ( ψ ⊗ µ ) ◦ ∆ o CS Theorem MB-Vogtmann (2020) envoy The map φ associated to M g , n and the map τ associated to Out( F n ) are mutually inverse elements under this group: τ = φ ⋆ − 1 φ = τ ⋆ − 1 23

• Characters, i.e. linear maps ψ : H → A which fulfill ψ ( I ) = I A form a group under the convolution product, ψ ⋆ µ = m ◦ ( ψ ⊗ µ ) ◦ ∆ Theorem MB-Vogtmann (2020) The map φ associated to M g , n and the map τ associated to Out( F n ) are mutually inverse elements under this group: τ = φ ⋆ − 1 φ = τ ⋆ − 1 • That means τ is the renormalized version of φ . 23

• Recall that χ ( M g , n ) is explicitly encoded by a TFT: � � � � χ ( M g , n ) 1 z 2 − 2 g − n = φ ( X ) = log e z (1+ x − e x ) dx √ n ! 2 π z R g , n 24

• Recall that χ ( M g , n ) is explicitly encoded by a TFT: � � � � χ ( M g , n ) 1 z 2 − 2 g − n = φ ( X ) = log e z (1+ x − e x ) dx √ n ! 2 π z R g , n • The duality between φ and τ implies that χ (Out( F n )) is encoded by the renormalization of the same TFT: � � � 1 e z (1+ x − e x )+ x 2 + T ( − ze x ) dx 0 = log √ 2 π z R where T ( z ) = � n ≥ 1 χ (Out( F n +1 )) z − n . 24

• Recall that χ ( M g , n ) is explicitly encoded by a TFT: � � � � χ ( M g , n ) 1 z 2 − 2 g − n = φ ( X ) = log e z (1+ x − e x ) dx √ n ! 2 π z R g , n • The duality between φ and τ implies that χ (Out( F n )) is encoded by the renormalization of the same TFT: � � � 1 e z (1+ x − e x )+ x 2 + T ( − ze x ) dx 0 = log √ 2 π z R where T ( z ) = � n ≥ 1 χ (Out( F n +1 )) z − n . renormalized TFT action 24

• Recall that χ ( M g , n ) is explicitly encoded by a TFT: � � � � χ ( M g , n ) 1 z 2 − 2 g − n = φ ( X ) = log e z (1+ x − e x ) dx √ n ! 2 π z R g , n • The duality between φ and τ implies that χ (Out( F n )) is encoded by the renormalization of the same TFT: � � � 1 e z (1+ x − e x )+ x 2 + T ( − ze x ) dx 0 = log √ 2 π z R where T ( z ) = � n ≥ 1 χ (Out( F n +1 )) z − n . • This TFT encodes the statement of Theorem 2 and gives an implicit encoding of the numbers χ (Out( F n )). 24

Outlook: The naive Euler characteristic 25

Outlook: The naive Euler characteristic The ‘naive’ Euler characteristic � ( − 1) k dim H k (Out F n ; Q ) χ (Out F n ) = � k is harder to analyse than the rational Euler characteristic. 25

Outlook: The naive Euler characteristic The ‘naive’ Euler characteristic � ( − 1) k dim H k (Out F n ; Q ) χ (Out F n ) = � k is harder to analyse than the rational Euler characteristic. � χ (Out F n ) = χ ( C σ ) � � σ � sum over conjugacy elements of finite order in Out F n and C σ is the centralizer corresponding σ Brown (1982) . 25