MAT 543, Fall 1999 Problem set 1

1. a. Classify all discrete submodules of ℂ. b. More generally, classify all torsion free commutative discrete subgroups of PSL(2, ℂ); that is, find a family of nonconjugate subgroups {G_{λ}} parameterized by λ in some set of Λ so that each torsion free commutative discrete subgroup of PSL(2, ℂ) is conjugate to precisely one G_{λ}. c. Classify all commutative discrete subgroups of PSL(2, ℂ).

2. Let τ ∈ ℍ^{2}. Can a torus T_{τ} carry a meromorphic function that is unbranched? Branched only over ∞?

3. Let [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ PSL(2, ℂ) and τ ∈ ℍ^{2}. The map C : z ↦ \frac{z}{cτ+d} is (induces) a conformal

map from T_{τ} to T_{\frac{aτ+b}{cτ+d}} and hence the map f ↦ f ∘ C is an isomorphism of K(T_{\frac{aτ+b}{cτ+d}}) onto K(T_{τ}).

4. Show that for each γ ∈ Γ = PSL(2, ℤ), there is a permutation σ_{γ} of the integers 1,2 and 3, such that

(e_{i} ∘ γ)γ′ = e_{σ_{γ}(i)}, i = 1, 2, 3.

Further, the map γ ↦ σ_{γ} is a surjective homomorphism from Γ to the permutation group on three letters with kernel the level two principal congruence subgroup Γ(2).

5. Describe a fundamental domain for the action of Γ(2) on ℍ^{2}. Prove that you indeed have a fundamental domain.

6. Study the action of SL(2, ℤ) on ℂ × ℍ^{2}.

7. a. Describe the group of conformal automorphisms of the torus T_{τ}, Aut(T_{τ}). b. Find the formula for each element of Aut(S_{t}) in terms of the plumbing parameters on S_{t}.

8. Let F be a family of meromorphic functions on a domain D. Assume that each f ∈ F omits three distinct values a, b and c ∈ ℂ̂. Show that F is normal.

9. Let D be a domain in ℂ̂. Let Maps(D) be the set of holomorphic maps from D to ℂ̂; that is, the field of meromorphic functions on D with the constant map whose value is ∞ adjoined. Let a ∈ ℂ̂ and f ∈ Maps(D). An a-point for f is a point z ∈ D with f(z) = a. The multiplicity of f at z is the local degree of f (restricted to a small neighborhood of z) at z. The multiplicity of a nonconstant f at each a point is a positive integer if a belongs to the range of f. Define the multiplicity of a point not in the range of f to be +∞. Define the multiplicity of a constant map appropriately. Let a_{1}, …, a_{n} be a collection of n distinct points in ℂ̂. Choose n positive integers or +∞: ν_{1}, …, ν_{n}. Let

F = {f ∈ Maps(D); the multiplicity of each a_{i} point is ≥ ν_{i}}.

Find necessary and sufficient conditions for F to be normal.

10. Let f be a self homeomorphism of the unit circle. Let F be the solution to the Dirichlet problem for the unit disc with boundary values f. Prove that F is a self homeomorphism of the closed unit disc.

11. Assume that u is hamonic for R_{1} < |z| < R_{2}. Show that there exist α and β ∈ ℂ such that

\frac{1}{2π} ∫_{0}^{2π} u(re^{ıθ})dθ = α log r + β,

for all R_{1} < r < R_{2}.