MATH32051
HYPERBOLIC GEOMETRY
20th Jan 2020
1.
(i) Recall that a straight line in R2 is given by the equation ax+by +c = 0, a,b, c ∈ R. Show that this equation can be written in the form.
βz + βz + = 0, β ∈ C, ∈ R.
Show that a vertical straight line in C has an equation of the form.
βz + βz + = 0, β, ∈ R. [4 marks]
(ii) Recall that the (Euclidean) circle in C with centre z0 ∈ C and radius r > 0 is given by the
equation |z - z0 | 2 = r2. Show that this equation can be written in the form.
zz + βz + βz + = 0, β ∈ C, ∈ R
and determine β, in terms of z0 ,r.
Show that the equation of a (Euclidean) circle in C with real centre z0 has the form.
zz + βz + βz + = 0, β, ∈ R [6 marks]
(iii) Consider the points -5 + 12i, 12 + 5i ∈ H. Find the equation of the geodesic (i.e. ind an equation of the form (1)) that passes through these two points.
This geodesic is a semi-circle. Determine its centre z0 and radius r. Hence write down the end-points of this geodesic.
Write down a M¨obius transformation of H that maps this geodesic to the imaginary axis.
The point z1 = (39+52i)/5 also lies on this geodesic. Briely explain how you would construct a M¨obius transformation that maps this geodesic to the imaginary axis and maps the point z1 to i. (You do not need explicitly calculate this M¨obius transformation; instead, your answer should explain how you would do it.) [8 marks]
(iv) Let 0 < a < b. Let σ be a path from ia to ib. Prove that lengthH (σ) ≥ logb/a with equality if, and only if, σ is the straight line along the imaginary axis from ia to ib. [8 marks]
(v) Consider the following statement.
Let H1 , H2 be two geodesics in H that do not intersect. Then there exists a unique geodesic in H that passes through both H1 and H2 at right-angles.
Suppose that H1 , H2 have distinct end-points on ∂H. Prove that, in this case, the above statement is true. (Hint: Without loss of generality you can assume that H1 is the imaginary axis. Which geodesics pass through H1 at right-angles?) [4 marks]
2.
(i) Recall that a M¨obius transformation of D is a transformation of the form.
(z) =
where α,β ∈ C and |α| 2 - |β| 2 > 0.
In each of the following cases, state whether the transformation is a M¨obius transformation of D or not, giving a brief reason for your answer:
(a) (z) = eiθ z, θ ∈ R, (b) (z) = . [4 marks]
(ii) Let
1 (z) = , 2 (z) = ∈ M¨ob(D)
be two M¨obius transformations of the Poincare disc D. (Here α1 ,α2 ,β1 ,β2 ∈ C and |α1 | 2 - |β1 | 2 > 0, |α2 | 2 - |β2 | 2 > 0.)
Show that the composition 1 2 is a M¨obius transformation of D. [8 marks]
(iii) Let Γ M¨ob(D) be a Fuchsian group. Briely outline a procedure which will generate a Dirichlet region for Γ .
Let 3 denote the rotation around the origin through 120 degrees anticlockwise; let 4 denote the rotation around the origin through 120 degrees clockwise. Let Γ = {id, 3 , 4 }.
Let p = 1/2 and determine the Dirichlet polygon D(p).
Sketch the resulting tessellation in D.
Determine the side-pairing transformations of D(p).
Sketch the corresponding tessellation in the upper half-plane H. [14 marks]
(iv) Let Γ = {id, 3 , 4 } be as in (iii) above. Give an example of a fundamental domain D for Γ
that cannot be of the form D(p) for any p ∈ D.
Sketch the resulting tessellation in D. [4 marks]
3. Throughout this question you may use the fact that coshdH (z, w) = 1 + . You may
also use the fact that sinπ/6 = 1/2.
(i) Recall that if A H then we deine AreaH (A) = llA .
Let Δ be a hyperbolic triangle with one ideal vertex and internal angles α,β,0. Prove that AreaH (A) = π - (α+β). (You may NOT assume the Gauss-Bonnet Theorem.) [8 marks]
(ii) Consider the diagram in Figure 1(i) below. Show that
sin θ = . (2)
[Hint: suppose the geodesic through 1 and ib is a semi-circle with centre x and radius r. Consider the (Euclidean) right-angled triangle with vertices at x, 0, ib and use the (Euclidean) Pythagoras Theorem.]
Now consider the hyperbolic triangle in H with vertices at (2 + √3)i, 0, 1 as illustrated in Figure 1(ii). What are the internal angles at 0 and 1? Use the Gauss-Bonnet Theorem and (2) to calculate the hyperbolic area of this triangle.
0 1
(2 + √3)i
0 1
Figure 1: See Q3(ii). [8 marks]
(iii) Let Δ be a right-angled hyperbolic triangle with one ideal vertex and internal angles α,0,π/2. Then Δ has one side with inite hyperbolic length; let a denote the hyperbolic length of this side.
Prove the angle of parallelism formula: cosha = 1/ sinα .
Is there a Euclidean analogue of this result? [8 marks]
(iv) Let Δ be a right-angled hyperbolic triangle with one ideal vertex and internal angles α,0,π/2.
Suppose that the side of Δ of inite hyperbolic length has hyperbolic length log(2 + √3). Calculate AreaH (Δ). [Hint: you can use the fact that cosha = (ea + e — a )/2.] [6 marks]
4.
(a) (i) Let E be an elliptic cycle with corresponding elliptic cycle transformation . What does it mean to say that E satisies the elliptic cycle condition?
Let P be a parabolic cycle with corresponding parabolic cycle transformation . What
does it mean to say that P satisies the parabolic cycle condition? [4 marks]
(ii) Consider the hyperbolic polygon as illustrated in Figure 2.
1
2
3 + 3i
0
Figure 2: See Q4(a)(ii).
Deine
1 (z) = z + 6, 2 (z) = z—3 (3)
Use Poincare’s Theorem to show that 1 and 2 generate a Fuchsian group Γ . Give a presentation of Γ in terms of generators and relations. Briely describe the quotient space H/Γ .
Show by explicit calculation that 1 , 2 , as deined in (3), satisfy the relation or relations that you have given in your presentation of Γ . [14 marks]
(b) (i) Let S = {a1 ,..., ak } be a inite set of symbols. Briely explain how to construct the free group on k generators, Fk .
(Your answer should include: a description of the elements of Fk , a description of the group operation, a description of the group identity, a description of how to ind the inverse of an element in Fk . You do not need to prove that the group operation is well-deined.) [4 marks]
(ii) Consider F2 , the free group on 2 generators a,b. List the 4 distinct words of length 1 and the 12 distinct words of length 2 in F2 .
How many distinct words of length n are there in F2 ? Justify your answer. [8 marks]