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MATH32051 HYPERBOLIC GEOMETRY

创建日期：2024-08-16 15:18:06

所属分类：数学mathematics

标签： MATH32051

HYPERBOLIC GEOMETRY

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MATH32051

HYPERBOLIC GEOMETRY

(i) Let

Prove that Γ is a subgroup of M¨ob(H).

(You may use facts from the course about composing M¨obius transformations and matrices without proof, provided that you state them clearly.) [10 marks]

(i) State, without proof, the Gauss-Bonnet Theorem for a hyperbolic polygon. [2 marks]

(ii) Consider the diagram below. Let e ≥ 1. Here, the semi-circle between ie and 1 is a geodesic. By drawing in the (Euclidean) centre and (Euclidean) radius of the semi-circle passing through 1 and i` and considering a (Euclidean) right-angled triangle, show that [6 marks]

(iii) Consider the hyperbolic quadrilateral P with vertices at ?1, 0, 1, i√ 3 below. Determine (in the form. aπ/b for suitable integers a, b) the hyperbolic area of P, giving reasons for your answer.

(You may use the fact that sin π/3 = √ 3/2 without proof.) [6 marks]

(i) Let γ1, γ2 ∈ M¨ob(H). What does it mean to say that γ1, γ2 are conjugate? [2 marks]

(ii) Recall that a M¨obius transformation γ ∈ M¨ob(H) is said to be parabolic if γ has a unique fixed point on ?H and no fixed points in H. The following result was proved in the course:

Let γ ∈ M¨ob(H) and suppose that γ = id. Then the following are equivalent:

(1) γ is parabolic

(2) τ (γ) = 4

(3) γ is conjugate to a translation z 7→ z + b

(4) γ is conjugate either to z 7→ z + 1 or to z 7→ z ? 1.

Prove that (1) ? (3). (You do not need to prove the other implications.) [8 marks]

(iii) Suppose that γ ∈ M¨ob(H) is parabolic. Is γ ?1 parabolic? Give either a proof or a counter-example. [4 marks]

(iv) Letγ1(z) = z + β, β ∈ R, β = 0, be a translation. Suppose that γ2 ∈ M¨ob(H) commutes with γ1 (so that γ1γ2(z) = γ2 γ1 (z) for all z). Show that if γ2 commutes with γ1 then γ2 is also a translation. [6 marks]

(i) Let Γ be a Fuchsian group acting on H. What does it mean to say that a set F ? H is a fundamental domain for Γ? [3 marks]

(ii) Briefly explain how to construct a Dirichlet region for a Fuchsian group Γ. (Your answer should include a statement of how to choose p, a definition of the terms [p, γ(p)], Lp(γ), Hp(γ) and D(p) that were defined in the course.) [5 marks]

(iii) Let z1 = x1 + iy1, z2 = x2 + iy2 ∈ H. It was proved in the course that the perpendicular bisector of [z1, z2], the arc of geodesic from z1 to z2, is given by {z ∈ H | dH(z, z1) = dH(z, z2)}. It was also shown in the course that this can be rewritten as

(You do not need to prove this.)

Use (?) to show that the perpendicular bisector of [1 + i, 4 n + 4n i] is

[6 marks]

(iv) Let Γ = {γn | γn(z) = 4n z, n ∈ Z}. Let p = 1 + i. Use the result of (iii) to calculate D(p). [4 marks]

(v) Let Γ be as in (iv). Is there a fundamental domain F for Γ with AreaH(F) < ∞? Briefly justify your answer. [4 marks]

(i) Let E = v0 → v1 → · · · → vn?1 be an elliptic cycle with corresponding elliptic cycle transformation γv0,s0. What does it mean to say that E satisfies the Elliptic Cycle Condition?

Let P = v0 → v1 → · · · → vn?1 be a parabolic cycle with corresponding parabolic cycle transformation γv0,s0. What does it mean to say that P satisfies the Parabolic Cycle Condition? [4 marks]

(ii) Consider the polygon P illustrated below with sides paired as indicated. (Here all 3 lines are geodesics. An extra vertex has been introduced at i/4 so that no side is paired with itself.)

Let k > 0. Let

Verify that γ1, γ2 pair the sides as indicated. [4 marks]

(iii) Determine the elliptic cycles and parabolic cycles. For each elliptic cycle, determine the elliptic cycle transformation and the angle sum along the elliptic cycle. For each parabolic cycle, determine the parabolic cycle transformation. [10 marks]

(iv) Determine the value of k for which γ1, γ2 generate a Fuchsian group Γ? In this case, give a presentation of Γ in terms of generators and relations. [10 marks]

(vi) For the Fuchsian group Γ determined in (iv), how many distinct words in Γ are there are of length 3? Briefly explain your reasoning.

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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