2 hours THE UNIVERSITY OF MANCHESTER.?? January 2017??:????:??

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1 hours MATH3051 THE UNIVERSITY OF MANCHESTER HYPERBOLIC GEOMETRY?? January 017??:????:?? Answer ALL FOUR questions in Section A (40 marks in all) and TWO of the THREE questions in Section B (30 marks each). If all three questions from Section B are attempted then credit will be given for the two best answers. Electronic calculators may be used, provided that they cannot store text. Notation: Throughout, H denotes the upper half-plane, H denotes the boundary of H, D denotes the Poincaré disc, and D denotes the boundary of D. 1 of 8 P.T.O.

2 SECTION A MATH3051 Answer ALL FOUR questions A1. (i) Let γ(z) = az + b, a, b, c, d R, ad bc > 0 cz + d be a Möbius transformation of H. Recall that z 0 H H is said to be a fixed point of γ if γ(z 0 ) = z 0. Suppose that c 0. Find the fixed points of γ. Explain how the number and location of fixed points can be used to classify Möbius transformations of H (in the case c 0). [4 marks] (ii) Let γ 1 (z) = 5z z + 1, γ (z) = 4z 3 z + 1. Determine whether γ 1, γ are hyperbolic, parabolic or elliptic Möbius transformations. A. Let be a hyperbolic right-angled triangle with sides of hyperbolic length a, b, c where c is the length of the side opposite the right-angle, as illustrated in Figure 1. c b a Figure 1: A hyperbolic right-angled triangle. See Question A. Prove the hyperbolic version of Pythagoras theorem: coshc = cosh a cosh b. (If you reduce the triangle to a special case then you should briefly justify why this is valid.) z w (You may use without proof the formula coshd H (z, w) = 1 + Im(z)Im(w).) of 8 P.T.O.

3 A3. (i) Let S = {a 1,...,a k } be a finite set of symbols and let w 1,...,w m be a finite set of words in symbols chosen from S S 1. Briefly explain how to construct the group Γ = a 1,..., a k w 1 = = w m = e. (Your answer should include: a description of the elements of Γ, a description of the group operation, a description of the group identity, and a brief explanation of how to find the inverse of an element in Γ. You do not need to prove that the group operation is well-defined.) (ii) Let Γ = a, b a 5 = b = (ab) = e. Show that ab = ba 4. Hence show that Γ contains precisely 10 elements. 3 of 8 P.T.O.

4 A4. (i) Consider the hyperbolic dodecagon as illustrated in Figure with all internal angles equal to π/6 and side-pairings as illustrated. Show that there is precisely one elliptic cycle and calculate the elliptic cycle transformation. Figure : Each internal angle is π/6 and the sides are paired as indicated. See Question A4. (ii) It can be shown from Poincaré s Theorem that the side-pairing transformations in this diagram generate a Fuchsian group Γ (you do not need to check this yourself). Sketch a picture of H/Γ. [ marks] 4 of 8 P.T.O.

5 SECTION B MATH3051 Answer TWO of the THREE questions B5. (i) Recall that the set of Möbius transformations of D is defined to be { Möb(D) = γ : D D γ(z) = αz + β } βz + ᾱ, α, β C, α β > 0. Let γ 1, γ Möb(D) and write γ 1 (z) = α 1z + β 1 β 1 z + α 1, γ (z) = α z + β β z + α. Show that the composition γ 1 γ is a Möbius transformation of D. Show that γ 1 1 is a Möbius transformation of D. [10 marks] (ii) Recall that if σ : [a, b] D is a parametrisation of a path in D then the hyperbolic length of σ is defined to be b length D (σ) = 1 σ(t) σ (t) dt. a How can the hyperbolic lengths of paths then be used to define a metric d D on D? (You do not need to prove that d D is a metric.) [ marks] (iii) Let a (0, 1) and consider the path σ along the imaginary axis that joins 0 and ia. Write down a parametrisation of σ. Hence show that ( ) 1 + a length D (σ) = log. 1 a (iv) Hence show that d D (0, ia) = log ( ) 1 + a. 1 a (v) Find the hyperbolic mid-point of the arc of geodesic in D between 0 and 4i/5. [4 marks] 5 of 8 P.T.O.

6 B6. (i) Let z 1 = x 1 +iy 1, z = x +iy H. In the course it was proved that the perpendicular bisector of [z 1, z ] is given by {z H d H (z, z 1 ) = d H (z, z )}. Show that the perpendicular bisector can also be written in the form {z H y z z 1 = y 1 z z }. (1) (You may use without proof the formula cosh d H (z, w) = 1 + z w Im(z)Im(w).) [4 marks] (ii) Let n Z. Show from (1) that the perpendicular bisector of [i, 9 n i] is the semi-circle with centre 0 and radius 3 n. (iii) Let Γ be a Fuchsian group. What does it mean to say that an open subset F H is a fundamental domain for Γ? Briefly outline an algorithm, discussed in lectures, that will construct a fundamental domain for Γ. (iv) Using this algorithm and part (ii) above, find a fundamental domain for the Fuchsian group Γ defined by Γ = {γ n γ n (z) = 9 n z, n Z}. Sketch the resulting tessellation of H. What does this tessellation look like in the Poincaré disc model D? (v) Let Γ be a Fuchsian group. A Fuchsian group Γ 1 is said to be conjugate to Γ if there exists a Möbius transformation g Möb(H) such that Γ 1 = g 1 Γg, that is Γ 1 = {g 1 γg γ Γ}. () Suppose that Γ 1 is conjugate to Γ and that g is as in (). Let F be a fundamental domain for Γ. Show that g 1 F is a fundamental domain for Γ 1. 6 of 8 P.T.O.

7 B7. (i) Recall that geodesics in H are either semi-circles with real centres or vertical straight lines in H and that they have equations of the form where α, β, γ R. αz z + βz + β z + γ = 0 (3) Consider the geodesic between 1, (1+i 3)/ and the geodesic between (1+i 3)/, (3+i 3)/. Note that both of these geodesics are contained in semi-circles with real centres. By finding equations of the form (3) or by using geometric intuition, find the centres and radii of these semi-circles. (ii) Let C 1, C be two Euclidean circles with centres c 1, c C and radii r 1, r, respectively. Suppose that C 1, C intersect as illustrated in Figure 3. C C 1 ψ c c 1 Figure 3: Circles C 1, C in C with centres c 1, c and radii r 1, r, respectively. See Question B7(ii). Recall that the Euclidean cosine rule states that in a Euclidean triangle with sides of (Euclidean) length a, b, c and with internal angle γ opposite the side c we have c = a +b ab cosγ. Use the Euclidean cosine rule to show that cosψ = c 1 c (r 1 + r ) r 1 r. (iii) Use the results of (i) and (ii) above to show that the angle θ 1 in Figure 4 below is equal to π/3. You may also assume that θ is equal to π/3 (you do not need to check this yourself).. [4 marks] 7 of 8 P.T.O.

8 (iv) Consider the hyperbolic triangle in Figure 4 below with sides paired as indicated. The sidepairing transformations are given by γ 1 (z) = 1 z +, γ (z) = z z 1. 1+i 3 θ 1 θ γ γ 1 3+i 3 1 Figure 4: A hyperbolic triangle with vertices at 1, Question B7(i), (iii), (iv). 1+i 3, 3+i 3 and internal angles θ 1, θ. See Use Poincaré s Theorem to show that γ 1, γ generate a Fuchsian group Γ. Show that there are two elliptic cycles and one parabolic cycle. Give a presentation of Γ in terms of generators and relations. [ 1+i (You may use, without proof, the fact that the hyperbolic midpoint of ] 3, 3+i 3 occurs at the point 1 + i.) [1 marks] END OF EXAMINATION PAPER 8 of 8