The Euler characteristic of Out( F n ) and the Hopf algebra of - - PowerPoint PPT Presentation
The Euler characteristic of Out( F n ) and the Hopf algebra of graphs Michael Borinsky, Nikhef September 18, ACPMS - Place: the Internet joint work with Karen Vogtmann arXiv:1907.03543 - to be published in Commentarii Mathematici Helvetici
Michael Borinsky, Nikhef September 18, ACPMS - Place: the Internet
joint work with Karen Vogtmann arXiv:1907.03543 - to be published in Commentarii Mathematici Helvetici
Teichm¨ uller space Tg MCG(Sg) mapping class group
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Teichm¨ uller space Tg MCG(Sg) mapping class group Harer Zagier (1986): χ(MCG(Sg)) = χ(Mg) = B2g 4g(g − 1)
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Teichm¨ uller space Tg MCG(Sg) mapping class group Harer Zagier (1986): χ(MCG(Sg)) = χ(Mg) = B2g 4g(g − 1) Culler-Vogtmann Outer space Xn Out(Fn)
This work: χ(Out(Fn)) = χ(Xn/ Out(Fn)) = . . .
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ρ : G → G such that ρ(x · y) = ρ(x) · ρ(y) for all x, y ∈ G
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ρ : G → G such that ρ(x · y) = ρ(x) · ρ(y) for all x, y ∈ G
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ρ : G → G such that ρ(x · y) = ρ(x) · ρ(y) for all x, y ∈ G
ρh :G → G, g → h−1gh for each h ∈ G.
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ρ : G → G such that ρ(x · y) = ρ(x) · ρ(y) for all x, y ∈ G
ρh :G → G, g → h−1gh for each h ∈ G.
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Fn = a1, . . . , an E.g. a1a−5
3 a2 ∈ Fn 3
Fn = a1, . . . , an E.g. a1a−5
3 a2 ∈ Fn
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Fn = a1, . . . , an E.g. a1a−5
3 a2 ∈ Fn
a1 → a1a2 a2 → a2 a3 → a3 . . . and a1 → a−1
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a2 → a2 a3 → a3 . . . and permutations of the letters.
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the mapping class group
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the mapping class group
MCG(Sg) := Out(π1(Sg))
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MCG(T2) = Out(π1(T2)) The group of homeomorphisms T2 → T2 up to an isotopy:
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How to study groups such as MCG(S) or Out(Fn)?
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How to study groups such as MCG(S) or Out(Fn)? Main idea Realize G as symmetries of some geometric object. Due to Stallings, Thurston, Gromov, . . . (1970-)
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Let S be a closed, connected and orientable surface.
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Let S be a closed, connected and orientable surface. ⇒ A point in Teichm¨ uller space T(S) is a pair, (X, µ)
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Let S be a closed, connected and orientable surface. ⇒ A point in Teichm¨ uller space T(S) is a pair, (X, µ)
MCG(S) acts on T(S) by composing to the marking: (X, µ) → (X, µ ◦ g−1) for some g ∈ MCG(S).
7 E.EE
Idea: Mimic previous construction for Out(Fn). Culler, Vogtmann (1986)
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Idea: Mimic previous construction for Out(Fn). Culler, Vogtmann (1986) Let Rn be the rose with n petals.
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Idea: Mimic previous construction for Out(Fn). Culler, Vogtmann (1986) Let Rn be the rose with n petals. ⇒ A point in Outer space On is a pair, (G, µ)
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Idea: Mimic previous construction for Out(Fn). Culler, Vogtmann (1986) Let Rn be the rose with n petals. ⇒ A point in Outer space On is a pair, (G, µ)
Out(Fn) acts on On by composing to the marking: (Γ, µ) → (Γ, µ ◦ g−1) for some g ∈ Out(Fn) = Out(π1(Rn)).
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Put picture of Outer space here
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Vogtmann 2008
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Scalar QFT ∼ Integrals over On / Out(Fn)
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Scalar QFT ∼ Integrals over On / Out(Fn) analogous to 2D Quantum gravity ∼ Integral overT(S)/ MCG(S)
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space of graphs.
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space of graphs.
curves.
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space of graphs.
curves.
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MCG(Sg) Out(Fn) acts freely and properly on Teichm¨ uller space T (Sg) Outer space On Quotient X/G Moduli space of curves Mg Moduli space of graphs Gn
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as On is contractible Culler, Vogtmann (1986).
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as On is contractible Culler, Vogtmann (1986). ⇒ Study Out(Fn) using Gn!
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as On is contractible Culler, Vogtmann (1986). ⇒ Study Out(Fn) using Gn!
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Consider the abelization map Fn → Zn.
Consider the abelization map Fn → Zn. ⇒ Induces a group homomorphism Out(Fn) → Out(Zn)
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Consider the abelization map Fn → Zn. ⇒ Induces a group homomorphism Out(Fn) → Out(Zn)
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Consider the abelization map Fn → Zn. ⇒ Induces a group homomorphism 1 → T n → Out(Fn) → Out(Zn)
→ 1
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Consider the abelization map Fn → Zn. ⇒ Induces a group homomorphism 1 → T n → Out(Fn) → Out(Zn)
→ 1
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Consider the abelization map Fn → Zn. ⇒ Induces a group homomorphism 1 → T n → Out(Fn) → Out(Zn)
→ 1
χ(Out(Fn)) = χ(GL(n, Z)) χ(T n)
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Consider the abelization map Fn → Zn. ⇒ Induces a group homomorphism 1 → T n → Out(Fn) → Out(Zn)
→ 1
χ(Out(Fn)) = χ(GL(n, Z))
χ(T n) n ≥ 3
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Consider the abelization map Fn → Zn. ⇒ Induces a group homomorphism 1 → T n → Out(Fn) → Out(Zn)
→ 1
χ(Out(Fn)) = χ(GL(n, Z))
χ(T n) n ≥ 3 ⇒ T n does not have finitely-generated homology for n ≥ 3 if χ(Out(Fn)) = 0.
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Conjecture Smillie-Vogtmann (1987) χ(Out(Fn)) = 0 for all n ≥ 2 and |χ(Out(Fn))| 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.
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Conjecture Smillie-Vogtmann (1987) χ(Out(Fn)) = 0 for all n ≥ 2 and |χ(Out(Fn))| 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.
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Conjecture Smillie-Vogtmann (1987) χ(Out(Fn)) = 0 for all n ≥ 2 and |χ(Out(Fn))| 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(H2(T n)) = ∞, which implies that T n does not have finitely-generated homology.
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Conjecture Smillie-Vogtmann (1987) χ(Out(Fn)) = 0 for all n ≥ 2 and |χ(Out(Fn))| 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(H2(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.
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Theorem A MB-Vogtmann (2019) χ(Out(Fn)) < 0 for all n ≥ 2
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Theorem A MB-Vogtmann (2019) χ(Out(Fn)) < 0 for all n ≥ 2 χ(Out(Fn)) ∼ − 1 √ 2π Γ(n − 3/2) log2 n as n → ∞.
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Theorem A MB-Vogtmann (2019) χ(Out(Fn)) < 0 for all n ≥ 2 χ(Out(Fn)) ∼ − 1 √ 2π Γ(n − 3/2) log2 n as n → ∞. which settles the initial conjecture by Smillie-Vogtmann (1987). Immediate questions:
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Theorem A MB-Vogtmann (2019) χ(Out(Fn)) < 0 for all n ≥ 2 χ(Out(Fn)) ∼ − 1 √ 2π Γ(n − 3/2) log2 n as n → ∞. which settles the initial conjecture by Smillie-Vogtmann (1987). Immediate questions: ⇒ Huge amount of unstable homology in odd dimensions.
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Theorem A MB-Vogtmann (2019) χ(Out(Fn)) < 0 for all n ≥ 2 χ(Out(Fn)) ∼ − 1 √ 2π Γ(n − 3/2) log2 n as n → ∞. which settles the initial conjecture by Smillie-Vogtmann (1987). Immediate questions: ⇒ Huge amount of unstable homology in odd dimensions.
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Theorem A MB-Vogtmann (2019) χ(Out(Fn)) < 0 for all n ≥ 2 χ(Out(Fn)) ∼ − 1 √ 2π Γ(n − 3/2) log2 n as n → ∞. which settles the initial conjecture by Smillie-Vogtmann (1987). Immediate questions: ⇒ Huge amount of unstable homology in odd dimensions.
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This Theorem A follows from an implicit expression for χ(Out(Fn)):
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This Theorem A follows from an implicit expression for χ(Out(Fn)): Theorem B MB-Vogtmann (2019) √ 2πe−NNN ∼
ak(−1)kΓ(N + 1/2 − k) as N → ∞ where
akzk = exp
n≥0
χ(Out(Fn+1))zn
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This Theorem A follows from an implicit expression for χ(Out(Fn)): Theorem B MB-Vogtmann (2019) √ 2πe−NNN ∼
ak(−1)kΓ(N + 1/2 − k) as N → ∞ where
akzk = exp
n≥0
χ(Out(Fn+1))zn ⇒ χ(Out(Fn)) are the coefficients of an asymptotic expansion.
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This Theorem A follows from an implicit expression for χ(Out(Fn)): Theorem B MB-Vogtmann (2019) √ 2πe−NNN ∼
ak(−1)kΓ(N + 1/2 − k) as N → ∞ where
akzk = exp
n≥0
χ(Out(Fn+1))zn ⇒ χ(Out(Fn)) are the coefficients of an asymptotic expansion.
Theorem B.
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This Theorem A follows from an implicit expression for χ(Out(Fn)): Theorem B MB-Vogtmann (2019) √ 2πe−NNN ∼
ak(−1)kΓ(N + 1/2 − k) as N → ∞ where
akzk = exp
n≥0
χ(Out(Fn+1))zn ⇒ χ(Out(Fn)) are the coefficients of an asymptotic expansion.
Theorem B.
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Similar result for the mapping class group/moduli space of curves:
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Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) χ(Mg) = χ(MCG(Sg)) = B2g 4g(g − 1) g ≥ 2
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Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) χ(Mg) = χ(MCG(Sg)) = B2g 4g(g − 1) g ≥ 2
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Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) χ(Mg) = χ(MCG(Sg)) = B2g 4g(g − 1) g ≥ 2
Penner (1988).
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Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) χ(Mg) = χ(MCG(Sg)) = B2g 4g(g − 1) g ≥ 2
Penner (1988).
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Similar result for the mapping class group/moduli space of curves: Theorem Harer-Zagier (1986) χ(Mg) = χ(MCG(Sg)) = B2g 4g(g − 1) g ≥ 2
Penner (1988).
⇒ Kontsevich’s proof served as a blueprint for χ(Out(Fn)).
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χ(Mg,n) n! z2−2g−n =
(−1)|VG | | Aut G| zχ(G).
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χ(Mg,n) n! z2−2g−n =
(−1)|VG | | Aut G| zχ(G).
by Penner (1986) based on ribbon graphs.
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χ(Mg,n) n! z2−2g−n =
(−1)|VG | | Aut G| zχ(G).
by Penner (1986) based on ribbon graphs.
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n
χ(Mg,n) n! z2−2g−n =
(−1)|VG | | Aut G| zχ(G).
by Penner (1986) based on ribbon graphs.
a ‘topological field theory’:
(−1)|VG | | Aut G| zχ(G) = log
√ 2πz
ez(1+x−ex)dx
ζ(−k) −k z−k
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χ(Mg,n) n! z2−2g−n =
(−1)|VG | | Aut G| zχ(G).
by Penner (1986) based on ribbon graphs.
a ‘topological field theory’:
(−1)|VG | | Aut G| zχ(G) = log
√ 2πz
ez(1+x−ex)dx
ζ(−k) −k z−k
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χ(Mg,n) n! z2−2g−n =
(−1)|VG | | Aut G| zχ(G).
by Penner (1986) based on ribbon graphs.
a ‘topological field theory’:
(−1)|VG | | Aut G| zχ(G) = log
√ 2πz
ez(1+x−ex)dx
ζ(−k) −k z−k
1 → π1(Sg,n) → Mg,n+1 → Mg,n → 1
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χ(Mg,n) n! z2−2g−n =
(−1)|VG | | Aut G| zχ(G).
by Penner (1986) based on ribbon graphs.
a ‘topological field theory’:
(−1)|VG | | Aut G| zχ(G) = log
√ 2πz
ez(1+x−ex)dx
ζ(−k) −k z−k
1 → π1(Sg,n) → Mg,n+1 → Mg,n → 1
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H = Q ∅
+ Q + Q + . . .
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H = Q ∅
+ Q + Q + . . .
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H = Q ∅
+ Q + Q + . . .
χ(Mg,n) n! z2−2g−n = φ(X) where X :=
G | Aut G|zχ(G) ∈ H[[z−1]] and φ : H → Q, G → (−1)|VG |
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H = Q ∅
+ Q + Q + . . .
χ(Mg,n) n! z2−2g−n = φ(X) where X :=
G | Aut G|zχ(G) ∈ H[[z−1]] and φ : H → Q, G → (−1)|VG | ⇒ φ is very simple and easy to handle via topological field theory.
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χ(Out(Fn+1))z−n = τ(X) with X as before and τ : H → Q, G →
(−1)|Ef | where the sum is over all forests (acyclic subgraphs) of G.
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χ(Out(Fn+1))z−n = τ(X) with X as before and τ : H → Q, G →
(−1)|Ef | where the sum is over all forests (acyclic subgraphs) of G. ⇒ Not directly approachable with a TFT...
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χ(Out(Fn+1))z−n = τ(X) with X as before and τ : H → Q, G →
(−1)|Ef | where the sum is over all forests (acyclic subgraphs) of G. ⇒ Not directly approachable with a TFT...
construction by Culler-Vogtmann (1986).
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m : H ⊗ H → H, G1 ⊗ G2 → G1 ⊎ G2 as multiplication, the empty graph ∅ associated with the neutral element I,
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m : H ⊗ H → H, G1 ⊗ G2 → G1 ⊎ G2 as multiplication, the empty graph ∅ associated with the neutral element I,
∆ : G →
bridgeless g
g ⊗ G/g, where the sum is over all bridgeless subgraphs,
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m : H ⊗ H → H, G1 ⊗ G2 → G1 ⊎ G2 as multiplication, the empty graph ∅ associated with the neutral element I,
∆ : G →
bridgeless g
g ⊗ G/g, where the sum is over all bridgeless subgraphs,
Kreimer (2009), which is closely related to the Hopf algebra
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ψ(I) = IA form a group under the convolution product, ψ ⋆ µ = m ◦ (ψ ⊗ µ) ◦ ∆
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ψ(I) = IA form a group under the convolution product, ψ ⋆ µ = m ◦ (ψ ⊗ µ) ◦ ∆ Theorem MB-Vogtmann (2020) The map φ associated to Mg,n and the map τ associated to Out(Fn) are mutually inverse elements under this group: τ = φ⋆−1 φ = τ ⋆−1
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envoy
ψ(I) = IA form a group under the convolution product, ψ ⋆ µ = m ◦ (ψ ⊗ µ) ◦ ∆ Theorem MB-Vogtmann (2020) The map φ associated to Mg,n and the map τ associated to Out(Fn) are mutually inverse elements under this group: τ = φ⋆−1 φ = τ ⋆−1
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χ(Mg,n) n! z2−2g−n = φ(X) = log
√ 2πz
ez(1+x−ex)dx
χ(Mg,n) n! z2−2g−n = φ(X) = log
√ 2πz
ez(1+x−ex)dx
encoded by the renormalization of the same TFT: 0 = log
√ 2πz
ez(1+x−ex)+ x
2 +T(−zex)dx
n≥1 χ(Out(Fn+1))z−n. 24
χ(Mg,n) n! z2−2g−n = φ(X) = log
√ 2πz
ez(1+x−ex)dx
encoded by the renormalization of the same TFT: 0 = log
√ 2πz
ez(1+x−ex)+ x
2 +T(−zex)dx
n≥1 χ(Out(Fn+1))z−n. 24
χ(Mg,n) n! z2−2g−n = φ(X) = log
√ 2πz
ez(1+x−ex)dx
encoded by the renormalization of the same TFT: 0 = log
√ 2πz
ez(1+x−ex)+ x
2 +T(−zex)dx
n≥1 χ(Out(Fn+1))z−n.
implicit encoding of the numbers χ(Out(Fn)).
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The ‘naive’ Euler characteristic
(−1)k dim Hk(Out Fn; Q) is harder to analyse than the rational Euler characteristic.
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The ‘naive’ Euler characteristic
(−1)k dim Hk(Out Fn; Q) is harder to analyse than the rational Euler characteristic.
χ(Cσ) sum over conjugacy elements of finite order in Out Fn and Cσ is the centralizer corresponding σ Brown (1982).
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The ‘naive’ Euler characteristic
(−1)k dim Hk(Out Fn; Q) is harder to analyse than the rational Euler characteristic.
χ(Cσ) sum over conjugacy elements of finite order in Out Fn and Cσ is the centralizer corresponding σ Brown (1982). ⇒ Preliminary investigations on Cσ indicate that lim
n→∞
χ(Out Fn) = c > 0
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Short summary:
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Short summary:
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Short summary:
Open questions:
unstable homology.
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Short summary:
Open questions:
unstable homology. Preliminary investigations into the χ(Out(Fn)) support this.
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Short summary:
Open questions:
unstable homology. Preliminary investigations into the χ(Out(Fn)) support this. What generates it?
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Short summary:
Open questions:
unstable homology. Preliminary investigations into the χ(Out(Fn)) support this. What generates it?
MCG(Sg) and Out(Fn). Obvious candidate: Koszul duality (?)
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Short summary:
Open questions:
unstable homology. Preliminary investigations into the χ(Out(Fn)) support this. What generates it?
MCG(Sg) and Out(Fn). Obvious candidate: Koszul duality (?)
groups and for finer invariants? For instance RAAGs or explicit homology groups.
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Generalize from Mg to Mg,n, the moduli space of surfaces of genus g and n punctures.
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Generalize from Mg to Mg,n, the moduli space of surfaces of genus g and n punctures. We can ‘forget one puncture’: MCG(Sg,n+1) → MCG(Sg,n)
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Generalize from Mg to Mg,n, the moduli space of surfaces of genus g and n punctures. We can ‘forget one puncture’: 1 → π1(Sg,n) → MCG(Sg,n+1) → MCG(Sg,n) → 1 ⇒ χ(MCG(Sg,n+1)) = χ(Mg,n+1) = χ(π1(Sg,n)) χ(Mg,n)
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Generalize from Mg to Mg,n, the moduli space of surfaces of genus g and n punctures. We can ‘forget one puncture’: 1 → π1(Sg,n) → MCG(Sg,n+1) → MCG(Sg,n) → 1 ⇒ χ(MCG(Sg,n+1)) = χ(Mg,n+1) = χ(π1(Sg,n))
χ(Mg,n)
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⇒ Ribbon graphs Penner (1986)
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⇒ Ribbon graphs Penner (1986) Every point in Mg,n can be associated with a ribbon graph Γ such that
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⇒ Ribbon graphs Penner (1986) Every point in Mg,n can be associated with a ribbon graph Γ such that
⇒ Γ can be interpreted as a surface of genus g with n punctures.
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n
⇒ Ribbon graphs Penner (1986) Every point in Mg,n can be associated with a ribbon graph Γ such that
⇒ Γ can be interpreted as a surface of genus g with n punctures.
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χ(Mg,n) =
(−1)dim(σ) | Stab(σ)|
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Sam
a
χ(Mg,n) =
(−1)dim(σ) | Stab(σ)|
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χ(Mg,n) =
(−1)dim(σ) | Stab(σ)| =
h0(∂Γ)=n χ(Γ)=2−2g−n
(−1)|VΓ| | Aut Γ|
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χ(Mg,n) =
(−1)dim(σ) | Stab(σ)| =
h0(∂Γ)=n χ(Γ)=2−2g−n
(−1)|VΓ| | Aut Γ| Used by Penner (1988) to calculate χ(Mg) with Matrix models.
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Kontsevich’s simplification:
χ(Mg,n) n! z2−2g−n
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Kontsevich’s simplification:
χ(Mg,n) n! z2−2g−n =
h0(∂Γ)=n χ(Γ)=2−2g−n
(−1)|VΓ| | Aut Γ| 1 n!zχ(Γ)
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Kontsevich’s simplification:
χ(Mg,n) n! z2−2g−n =
h0(∂Γ)=n χ(Γ)=2−2g−n
(−1)|VΓ| | Aut Γ| 1 n!zχ(Γ) =
(−1)|VG | | Aut G| zχ(G)
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Kontsevich’s simplification:
χ(Mg,n) n! z2−2g−n =
h0(∂Γ)=n χ(Γ)=2−2g−n
(−1)|VΓ| | Aut Γ| 1 n!zχ(Γ) =
(−1)|VG | | Aut G| zχ(G) This is the perturbative series of a simple TFT: = log
√ 2πz
ez(1+x−ex)dx
Kontsevich’s simplification:
χ(Mg,n) n! z2−2g−n =
h0(∂Γ)=n χ(Γ)=2−2g−n
(−1)|VΓ| | Aut Γ| 1 n!zχ(Γ) =
(−1)|VG | | Aut G| zχ(G) This is the perturbative series of a simple TFT: = log
√ 2πz
ez(1+x−ex)dx
=
ζ(−k) −k z−k
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2−2g−n=k
χ(Mg,n) n! = Bk+1 k(k + 1)
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2−2g−n=k
χ(Mg,n) n! = Bk+1 k(k + 1) ⇒ recover Harer-Zagier formula using the identity χ(Mg,n+1) = (2 − 2g − n)χ(Mg,n)
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Generalize from Out(Fn) to An,s and from On to On,s, Outer space of graphs of rank n and s legs. Contant, Kassabov, Vogtmann (2011)
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Generalize from Out(Fn) to An,s and from On to On,s, Outer space of graphs of rank n and s legs. Contant, Kassabov, Vogtmann (2011) Forgetting a leg gives the short exact sequence of groups 1 → Fn → An,s → An,s−1 → 1
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c Q
c On
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⇒ graphs with a forest Smillie-Vogtmann (1987):
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⇒ graphs with a forest Smillie-Vogtmann (1987): A point in Gn,s can be associated with a pair of a graph G and a forest f ⊂ G.
33
⇒ graphs with a forest Smillie-Vogtmann (1987): A point in Gn,s can be associated with a pair of a graph G and a forest f ⊂ G. (G, f )
33
χ(An,s) =
(−1)dim(σ) | Stab(σ)|
34
χ(An,s) =
(−1)dim(σ) | Stab(σ)|
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sum
χ(An,s) =
(−1)dim(σ) | Stab(σ)| =
with s legs rank(π1(G))=n
(−1)|Ef | | Aut G|
34
Renormalized TFT interpretation MB-Vogtmann (2019): χ(An,s) =
with s legs rank(π1(G))=n
1 | Aut G|
(−1)|Ef |
35
Renormalized TFT interpretation MB-Vogtmann (2019): χ(An,s) =
with s legs rank(π1(G))=n
1 | Aut G|
(−1)|Ef |
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Renormalized TFT interpretation MB-Vogtmann (2019): χ(An,s) =
with s legs rank(π1(G))=n
1 | Aut G|
(−1)|Ef |
τ fulfills the identities τ(∅) = 1 and
g bridgeless
τ(g)(−1)|EG/g| = 0 for all G = ∅
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Renormalized TFT interpretation MB-Vogtmann (2019): χ(An,s) =
with s legs rank(π1(G))=n
1 | Aut G|
(−1)|Ef |
τ fulfills the identities τ(∅) = 1 and
g bridgeless
τ(g)(−1)|EG/g| = 0 for all G = ∅ ⇒ τ is an inverse of a character in a Connes-Kreimer-type renormalization Hopf algebra. Connes-Kreimer (2001)
35
Renormalized TFT interpretation MB-Vogtmann (2019): χ(An,s) =
with s legs rank(π1(G))=n
1 | Aut G|
(−1)|Ef |
τ fulfills the identities τ(∅) = 1 and
g bridgeless
τ(g)(−1)|EG/g| = 0 for all G = ∅ ⇒ τ is an inverse of a character in a Connes-Kreimer-type renormalization Hopf algebra. Connes-Kreimer (2001) The group invariants χ(An,s) are encoded in a renormalized TFT.
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Let T(z, x) =
χ(An,s)z1−n xs s!
36
Let T(z, x) =
χ(An,s)z1−n xs s! then 1 = 1 √ 2πz
eT(z,x)dx
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Let T(z, x) =
χ(An,s)z1−n xs s! then 1 = 1 √ 2πz
eT(z,x)dx Using the short exact sequence, 1 → Fn → An,s → An,s−1 → 1 results in the action 1 = 1 √ 2πz
ez(1+x−ex)+ x
2 +T(−zex)dx
where T(z) =
n≥1 χ(Out(Fn+1))z−n. 36
Let T(z, x) =
χ(An,s)z1−n xs s! then 1 = 1 √ 2πz
eT(z,x)dx Using the short exact sequence, 1 → Fn → An,s → An,s−1 → 1 results in the action 1 = 1 √ 2πz
ez(1+x−ex)+ x
2 +T(−zex)dx
where T(z) =
n≥1 χ(Out(Fn+1))z−n.
This gives the implicit result in Theorem B.
36