Déformations géométriques de sous-groupes fermés

GG(G)
GG(G)
(Hn)n∈NG
H G
x∈H(xn)n∈NG x
n xn∈Hn
(nk)k∈N(xnk)k∈N
x xnk∈Hnkk x ∈H

RnRn
GL(n, R)
Rn
ZnGL(n, Z)Rn
GL(n, R)/GL(n, Z)
G(G)
§5
G(G)
G
G
{0}∪{1
n, n ∈N\{0}} → G(Z)1 n7→ nZ07→ {0}
Z
{0}nZ
G(Z)
G(Z)G(R/Z)
[0,∞]→ G(R) 0 7→ Rα7→ αZ∞ 7→ {0}
RαZ{0}
G(R)
RoR x·y=exy
G( )

R o {0}=RoR{0}
{0}o R
G( )
G(R2)R2
S44
3
R2o R R R2t·(x, y)=(x+ty, y)
GL(3,R)
1
R×Z
R×Z R ×Z
G(R×Z)H=Z(1
α,0) + Z(β α, n)α∈]0,∞[β∈R/Z
n∈N\{0} G(R×Z)
A⊂R2
(An)n∈N\{0}
CnAnR/Z(Cn)n∈N\{0}
CnQ/ZAnCn
CnA Ap0
n p φ(p
n)n p φ
G(R×Z) (Cn)n∈N\{0}
A

1
n= 3
α= 0
α=∞
α
n= 2 n= 1
n= 4
G(R×Z)
R×Z
G(R×Z)
M
p M spM
SnHn
R
−1
Rn
M

M
M0Rnn>1
n>2Hn
R
Rk16k6n−1
gθg
={X∈g:θX =X}p={X∈g:θX =−X}
G
K
M
G= Isom0(M)M
M
x0∈M K x0M
G/K →M
gK 7→ g·x0.
G/K
G
σ G dσe
gG K G (Gσ)0⊂K⊂Gσ
G G/K G/K
G G
Isom0(G/K)G/K
M
G/K G
G= SL(n, R)K= SO(n, R)
G G g
n K
gB(X, Y ) = 2ntr(XY )
θ:g→gX7→ −t X G
G/K
G
K