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) Park City, July 2012 1 / 11 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 2 / 11 � � � � � 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. 3 / 11 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 4 / 11 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. Park City, July 2012 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}. Park City, July 2012 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 5 / 11 Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (4/4) Park City, July 2012 6 / 11 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) Park City, July 2012 7 / 11 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.] Park City, July 2012 9 / 11 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 8 / 11 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 11 / 11 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) Park City, July 2012 10 / 11