Recall:
cochain
Some arithmetic groups
that do not act on the circle
→R
Hb∗ (Γ ; R): require all cochains to be bdd funcs on Γ n .
http://people.uleth.ca/~dave.morris
Dave.Morris@uleth.ca
Example
H 0 (Γ ; R) = R = Hb0 (Γ ; R).
Lecture 4
Intro to bounded cohomology
(used to prove actions have a fixed point)
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Example
Spse Γ acts on circle. I.e., Γ ⊂ Homeo+ (R/Z).
� ∈ Homeo+ (R).
Each g ∈ Γ lifts to g
�
�
Not unique: g(t)
= g(t)
+ n, ∃n ∈ Z.
�
Can choose g(0)
∈ [0, 1).
�
�
�
�
� h(0)
Let c(g, h) = g
− gh(0)
∈ Z.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
Park City, July 2012
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�
�
�
�
� h(0)
c(g, h) = g
− gh(0)
Proof.
(⇐) Wolog fixed point is 0.
�
Then g(0)
= 0, so c(g, h) = 0 for all g, h.
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Proposition (Ghys)
[c] = 0 in Hb2 (Γ ; Z) ⇐⇒ Γ has a fixed point in S 1 .
(⇒) c(g, h) = ϕ(gh)−ϕ(g)−ϕ(h), ∃ bdd ϕ : Γ → Z.
�
�
Let g(x)
= g(x)
+ ϕ(g), so
� = gh,
� so Γ� is a lift of Γ to Homeo+ (R), and
�h
g
�
�
|g(0)|
≤ |g(0)|
+ |ϕ(g)| ≤ 1 + �ϕ�∞ .
Γ�-orbit of 0 is bdd subset of R, so has a supremum,
which is fixed pt of Γ�; img in S 1 is fixed pt of Γ .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
Park City, July 2012
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Theorem (Burger-Monod)
Corollary
Hb2 (Γ ; Z) = 0 ⇒ every action of Γ on S 1 has fixed pt.
Exercise
Hb2 (Γ ; R) = 0, H 1 (Γ ; R) = 0, Γ is finitely generated
�⇒ every action of Γ on S 1 has a finite orbit.
Theorem (Burger-Monod)
Comparison map Hb2 (Γ ; R) → H 2 (Γ ; R) is injective
if Γ is large arith group.
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Interested in Hb2 (Γ ) – applies to actions on the circle.
[c] = 0 in Hb2 (Γ ; Z) ⇐⇒ Γ has a fixed point in S 1 .
So [c] ∈ Hb2 (Γ ; Z). The bdd Euler class of the action.
Well defined: independent of basepoint “0”, etc.
Corollary (Ghys, Burger-Monod)
Γ = large arith group
and H 2 (Γ ; R) = 0
�⇒ every action of Γ on S 1 has a finite orbit.
Hb1 (Γ ; R) = { bounded homos c : Γ → R } = {0}.
Proposition (Ghys)
c(g, h) ∈ {0, 1}.
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H 1 (Γ ; R) = { homomorphisms c : Γ → R }
Bounded Euler class
Exercise
c is a 2-cocycle:
c(h, k) − c(gh, k) + c(g, hk) − c(g, h) = 0
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
H ∗ (Γ ; R)
Definition (bounded cohomology)
University of Lethbridge, Alberta, Canada
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
c: Γ
n
coboundary δ : C n (Γ ) → C n+1 (Γ )
Z n (Γ ; R)
ker δn
H n (Γ ; R) = n
=
B (Γ ; R)
Im δn−1
Dave Witte Morris
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
group cohomology
Comparison map Hb2 (Γ ; R) → H 2 (Γ ; R) is injective
if Γ is large arith group.
Kernel of the comparison map
Let c ∈ Zb2 (Γ ; R). Assume [c] = 0 in H 2 (Γ ; R).
I.e., c = δα, ∃α ∈ C 1 (Γ ). So
|α(gh) − α(g) − α(h)| = |δα(g, h)| ≤ �c�∞ is bdd.
α is almost a homo — a quasimorphism.
Exercise
Quasimorphisms(Γ , R)
.
NearHom(Γ , R)
NearHom(Γ , R) = { α : Γ → R | bdd dist from homo }
Kernel of Hb2 (Γ ) → H 2 (Γ ) is
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Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
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Exercise
Kernel of
Hb2 (Γ )
Example:
Quasimorphisms(Γ , R)
→ H (Γ ) is
.
NearHom(Γ , R)
2
Hb2 (F2 ) is infinite-dimensional.
Proof. Construct lots of quasimorphisms
(not homos).
Homo ϕa (x) = the (signed) # occurrences of a in x.
E.g., ϕa (a2 ba3 b−3 a−7 b2 ) = 2 + 3 − 7 = − 2.
Every homo F2 → R is a linear comb of ϕa and ϕb .
ϕab (x) = # occurrences of ab in x (reduced)
E.g., ϕab (a2 ba3 b−3 a−7 b2 ) = 1 − 1 = 0.
Exercise: 1) ϕw is a quasimorphism, ∀ reduced w.
2) ϕak is not within bdd distance of lin span of
{ϕb , ϕa , ϕak+1 , ϕak+2 , ϕak+3 , . . .}.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
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Exercises
[Hint: Short exact sequence 0 → Z → R → T → 0 yields
δ
long exact sequence Hb1 (Γ ; T) → Hb2 (Γ ; Z) → Hb2 (Γ ; R).]
2) Every quasimorphism is bounded on the set of
commutators {x −1 y −1 xy}.
3) SL(3, Z) has no unbounded quasimorphisms.
[Hint: Use the fact that it is boundedly gen’d by elementary mats.]
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D. W. Morris: Can lattices in SL(n, R) act on the
circle?, in Geometry, Rigidity, and Group Actions,
University of Chicago Press, Chicago, 2011.
http://arxiv.org/abs/0811.0051
D. Witte: Products of similar matrices. Proc.
Amer. Math. Soc. 126 (1998), no. 4, 1005–1015.
http://dx.doi.org/10.1090/
S0002-9939-98-04368-8
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
Exercise: Spse ϕ : Γ → R unbdd quasimorphism.
Stabilize: let ϕ(g) = lim ϕ(g n )/n.
ϕ is unbounded quasimorphism.
ϕ(h−1 gh) = ϕ(g).
{ g ∈ Γ | ϕ(g) > C } is normal semigroup.
Open Problem.
For Γ = SL(3, Z):
Every normal semigroup in Γ is a subgroup.
∀g ∈ Γ , e is a product of conjugates of g.
�
(nonempty)
bi-invariant partial order on Γ .
All are equivalent.
($100 for solution)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
Park City, July 2012
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Further reading
1) Hb2 (Γ ; R) = 0, H 1 (Γ ; R) = 0, Γ is finitely gen’d
�⇒ every action of Γ on S 1 has a finite orbit.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
Example: homomorphism ϕ : Γ → R
�⇒ { g ∈ Γ | ϕ(g) > 0 } is normal semigroup.
Park City, July 2012
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M. Gromov: Volume and bounded cohomology.
Publ. Math. IHES 56 (1982) 5–99.
http://archive.numdam.org/article/
PMIHES_1982__56__5_0.pdf
N. Monod: An invitation to bounded
cohomology. Proc. Internat. Congress Math.,
Madrid, Spain, 2006.
http://www.mathunion.org/ICM/ICM2006.2/
Main/icm2006.2.1183.1212.ocr.pdf
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4)
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