Hyperbolicite et logique du premier

ΓG
Th(G) = Th(Γ) G

ΓG
Th(G) = Th(Γ) G
Γ0Γ
X= (Γ,{Xv}v∈V(Γ),{Xe}e∈E(Γ))
Xe→XvXe→Xwe= (v, w)
X XvXe
Xe×[0,1]
X1, X2
ij:§1→XjX1X2
G= (Γ,{Gv}v∈V(Γ),{Ge}e∈E(Γ))
Xe→XvXe→Xwe= (v, w)
X= (Γ,{Xv}v∈V(Γ),{Xe}e∈E(Γ))Xv
π1(Xv) = GvXeπ1(Xe) = GeXe→Xv
Ge→Gv
π1(X)
ΓGe'Z
π1(G) = G1∗ZG2
G= (Γ,{Gv}v∈V(Γ),{Ge}e∈E(Γ))
v Gv
e7→ [iv(Ge)] []
v
G0G
(G, G0, r)r:G→G0
G= (Γ,{Gv}v∈V(Γ),{Ge}e∈E(Γ))
V(Γ) = Vs(Γ) tVr(Γ) v∈Vs(Γ)
−2
ΓVsVr
G0Gvv∈V(Γ)
v∈Vs(Γ) r(Gv)
H G H
H H ≤G0≤ · · · ≤ Gm
•G0=H∗S1∗. . . ∗Sq∗FkSj
F
•i ri:Gi→Gi−1(Gi, Gi−1, ri)

2m∆:[−m, m]×
§1[i, i + 1] × §1∆−1({i} × §1)
∆({0}ק1)
∆([i, i + 1] × {∗})
G
•
•
•λ m > 0ρ
Hρ
2m HρHρλ
GG
G
H G
ρ λ
>2
GΓGΓG=hx|
Σ(x)iG→ΓγΣ
Γ Σ
Σ0
G→Γ
G|=∃x1. . . xrΣ0(x1, . . . , xr)=1
Γ Γ
Σ
G
G→Γ
(G, G0, r)
G
Γ
u1, . . . , urG h :G→Γ

•h
•σ∈Mod(G)h◦σ ui
GZ
G
G
Z
G Z
G
X G =π1(X)
X1, . . . , Xr
GZ
σ
R σ|R
S
σ(S) = Sγ
GΓ
GΓ
Zk
Z Z2
Λ
f:G→G
ΓG
Th(G) = Th(Γ) GΓ
ρ:G→G
hs1, . . . , sm|ΣG(s1, . . . , sm)i
G H f :G→H

m(x1, . . . , xm)H
ΣG(x)=1
ΛG(x, y)
f, f0x, y
f
f0
{qi:G→Qi|i= 1, . . . , s}
G→Γqi
ui(s1, . . . , sm)qi
GΓf:G→
Γf0:G→Γf=f◦σ σ ∈Mod(G)
f0(ui)=1 i f(s) = x f0(s) = yΛG(x, y)
f:G→Γ, s 7→ x f0:G→Γ, s 7→ y
ΛG(x, y)f0(ui)=1 i
ΓxΣG(x) = 1 y
ΣG(y) = 1 ΛG(x, y)i= 1, . . . , i =q
ui(y)=1
GΓ
G x =s
f0:G→G
ui
G ρ :G→G
r:G→G0(G, G0, r)
ρ
f:S1→S2
f(S1)S2
f(S1)S2
S S2f(S1)
S2S1
Σ Σ
ρ