Introduction à la stabilité en homologie des groupes discrets

K
K

(Gn)n∈Nin:Gn→
Gn+1
i∈NN∈N
(in)∗:Hi(Gn)→Hi(Gn+1)Z
n≥N
Hst
i((Gk)) := Hi(colim
kGk)'colim kHi(Gk)'Hi(Gn).
(C,⊕,0)

0AC
Gn= AutC(A⊕n)Gn= AutC(M⊕A⊕n)
MCin:Gn→Gn+1
u7→ u⊕(0 →A)A⊕nA⊕n⊕n A⊕n+1 A⊕n⊕A
Σn+1
Gnin
lim1
k p
GL∞(k) = colim
n∈NGLn(k)
GLn(k)Z/p
H∗(GLn(k); Z/p)n
n≤2

k Hi(Gln(k); Z/p) 0 < i < n k
F2
Z/p k
K
A K
A
X
E
π1(X)X+X→X+
π1π1(X)π1(X)/N
π1(X)
GL(A)
E(A)
GL(A)K A
n > 0Kn(A) = πn(BGL(A)+)n= 1 n= 2
K1(A)GL(A)H1
K2(A)
A GL(A)H2(E(A))
K BGLn(A)
K

K
K0K1Z
K
K
(Gn)
in:Gn→Gn+1 n GnMn
GnMn→Resin(Mn+1)
Hk(Gn;Mn)→Hk(Gn+1;Mn+1)in
n k Gn
A⊕n
A
A