Hyperbolic manifolds, discrete groups and ergodic theory

Math 277 – Fall 1996 – Berkeley C. McMullen

Homework 2 Due 12 November 96

Attempt as many problems as you wish. Cite references or collaborators if used.

1. Let K be a number field (a finite extension of ℚ), and let G ⊂ GL_{n}(K) be a finitely generated group of matrices with entries in K. Prove that G contains a torsion-free subgroup H with index [G : H] < ∞.

(Hint: using the fact that K is a finite-dimensional vector space over ℚ, reduce to the case K = ℚ; then use the fact that G is contained in GL_{n}(ℤ[1/q]) for some q, by virtue of finite generation.)

2. Let Γ ⊂ Isom^{+}(ℍ) be a discrete group. Show that its limit set Λ ⊂ S_{∞}^{1} is either:

(a) a finite set of 0, 1 or 2 points;

(b) a Cantor set; or

(c) the full circle S_{∞}^{1}.

(A Cantor set is a compact, totally disconnected space without isolated points.)

3. Find the isoperimetric constant γ of a degree d regular tree. (By definition γ = inf |∂V |/|V | where the inf is taken over all finite sets of vertices.)

4. (a) Show that using both horocycle flows, you can move any vector in T_{1}ℍ to any other in 4 steps. That is, y = h_{+}^{a}h_{−}^{b}h_{+}^{c}h_{−}^{d}(x) for appropriate a, b, c, d ∈ ℝ.

(b) Show there are pairs of vectors for which 3 steps do not suffice.

(c) Let Λ, Λ′ ⊂ ℝ^{2} be lattices with area(ℝ/Λ) = area(ℝ/Λ). Show there exists a bijection f : Λ → Λ′ moving points a bounded distance; that is, such that sup_{x} d(f(x), x) < ∞.

(Hint: first prove it when Λ = ℤ ⊕ iyℤ and Λ′ = ℤ ⊕ (x + iy)ℤ. Then use the fact that shears generate SL_{2}ℝ as in (a).)

5. Let Y be a noncompact hyperbolic surface of finite volume. Prove that there is no divergent horocycle: for any v ∈ T_{1}Y, there is a compact set K such that for all S > 0, h^{s}(v) ∈ K for some s ≥ S.

6. Let Y be a compact hyperbolic surface, and let v : [0, 1] → T_{1}Y be a continuous path, not entirely contained in a single orbit of the geodesic flow.

Prove there exists an x ∈ [0, 1] such that the geodesic γ(t) through v(x) has a triple point: that is, such that the projection of γ(t) to Y passes three times through the same point.

7. Let p be a prime and let Γ(p) = {A ∈ P SL_{2}ℤ : A ≡ ±I mod p}. Let Y (p) = ℍ/Γ(p). Compute the genus and number of punctures of Y (p), p > 1.

8. Let G = ℤ ⋉ ℝ, where the action of ℤ on ℝ is generated by x ↦ λx, λ > 1.

(a) Show that for any unitary representation of G without ℝ-invariant vectors, the ℤ-action is mixing.

(b) Let T ∈ SL_{2}ℤ be a hyperbolic toral automorphism, determining a measure-preserving action T : X → X where X = ℝ^{2}/ℤ^{2}. Let h^{t} : X → X be the flow by translation along the expanding foliation of T.

Show the action of T and h^{t} together generate a representation of G on L_{0}^{2}(X).

(c) Show h^{t} has no invariant vectors, and conclude that T is mixing.