Groupes libres et développements de Magnus

G
Aut(G)
IAab
nH1(IAn)
IAab
n

G G
xy:= y−1xy
y
x:= yxy−1
x y G
[x, y] := xyx−1y−1.
G
x y z
G
ϕ:G−→ H
ϕ([x, y]) = [ϕx, ϕy],
ϕ(xy) = (ϕx)φy.
•[x, x] = 1
•[x, y]−1= [y, x]
•[x, yz] = [x, y] (y[x, z]) ,
•[[x, y],y z]·[[y, z],z x]·[[z, x],x y] = 1
•[x, y−1], z−1x·[z, x−1], y−1z·[y, z−1], x−1y= 1
A B G [A, B]G
[a, b] (a, b)∈A×B
A B
[A, B] = [B, A].
G[G, G]
G G/[G, G]
G Gab

GΓ(G) ΓG
ΓG
Γ1:= G,
Γk+1 := [G, Γk].
A B C
G
•[A, [B, C]],
•[B, [C, A]],
•[C, [A, B]].
N(A)A G
G A
G/N N
A, B C
Aut(G)G
GoAut(G).
G×Aut(G)
(g, σ)·(h, τ) := (gσ(h), στ).
GAut(G)G×1 1×Aut(G)GoAut(G)
[σ, g] = σ(g)g−1.
[Aut(G), G]⊆G

G G
N
G=N1⊇ · · · ⊇ Ni⊇ · · ·
G
[Ni, Nj]⊆Ni+j.
G=N1
[G, Ni]⊆Ni+1 ⊆NiNi
G
Ni⊇Γi(G)
i>1
ΓG
[Γi,Γj]⊆Γi+j.
i j
i= 0 G=
Γ1i−1j
[Γi,Γj] = [[Γi−1, G],Γj]⊆ N ([Γi−1,[G, Γj]] ∪[G, [Γi−1,Γj]])
⊆ N ([Γi−1,Γj+1]∪[G, Γi−1+j])
⊆ N (Γi+j∪Γi+j) = Γi+j G.
G
Di,k:= G∩1+(IkG)i,
IkGkG
k
Aut(G)
Aut(G)
Ak(G) := ker (Aut(G)−→ Aut(G/Γk+1(G))) .

Aut(G)−→ Aut(G/Γk+1(G))
ϕ
G ϕ ϕ−1Γi(G)G
ϕ G/Γk+1(G)
AkG/Γk+1(G)
IAG:= A1.
G
k, l >1
•[Ak,Γl]⊆Γk+l,
•[Ak,Al]⊆ Ak+l
GoAut(G)
A
A1=IAG
Ak={σ∈Aut(G)|[σ, G]⊆Γk+1(G).}
l= 1
l
[[Ak,Al], G]⊆Γk+l+1.
Ak(G) Γk(IAG)
∀k, Ak= Γk(IAG).
G=N1⊇ · · · ⊇ Nk⊇ · · ·
G[G, Ni]⊆NiNi
GLi(N) := Ni/Ni+1
i > 1
[Ni, Ni]⊆N2i⊆Ni+1.