Side parameter for the torus with a single cone point

Transcription

1 Side parameter for the torus with a single cone point Hirotaka Akiyoshi Osaka City University A Satellite Conference of Seoul ICM 2014 Knots and Low Dimensional Manifolds Aug 22, 2014 BEXCO Convention & Exhibition Center II, Busan, Korea

2 Goal of the project By the 1980s, Jorgensen characterized the combinatorial structures of the Ford domains of quasifuchsian once-punctured torus groups. (cf. A-Sakuma-Wada-Yamashita, LNM 1909, Springer) 1

3 Goal of the project By the 1980s, Jorgensen characterized the combinatorial structures of the Ford domains of quasifuchsian once-punctured torus groups. (cf. A-Sakuma-Wada-Yamashita, LNM 1909, Springer) Generalize this to the torus with a single cone point! cf. T. Jorgensen (1977): Compact 3-manifolds of constant negative curvature fibering over the circle 1

4 Torus with a single cone point We assume the cone angle θ [0, 2π). θ = 0 puncture (classical case) A hyperbolic structure on T θ is a metric s.t. locally isometric to H 2 on T θ {v} a nbd of v is isometric to (to a cusp for θ = 0) 2

5 The cone manifold M θ = T θ ( 1, 1) A hyperbolic structure on M θ is a metric s.t. locally isometric to H 3 on M θ Σ a nbd of q Σ is isometric to (to a cusp for θ = 0) 3

6 M 0 as the covering space of a torus bundle Let σ be the hyp str on M 0 = T 0 ( 1, 1) induced from the infinite cyclic covering of a hyp T 0 -bundle. [Thurston, Double Limit Theorem] σ QF for QF = {q.f. hyp str on M 0 }. [Ahlfors-Bers Theory] QF = Teich(T 0 ) Teich(T 0 ) = H 2 H 2. [Minsky, Solution of Thurston s ELC for T 0 ] QF = H 2 H 2 diag( H 2 ). 4

7 Quasifuchsian space for T 0 QF = {quasifuchsian hyp str on M 0 } = H 2 H 2 QF = H 2 H 2 diag( H 2 ) All hyp str σ corresponding to hyp T 0 -bdles are contained in H 2 H 2 diag( H 2 ). 5

8 Expectation for the space of cone hyp str T : the torus, v T, T 0 = T {v}, M = T ( 1, 1), Σ = {v} ( 1, 1) A naive expectation for the space of cone hyp str H on (M, Σ, θ) (0 θ < 2π) is as follows: 6

9 Expectation for the space of cone hyp str T : the torus, v T, T 0 = T {v}, M = T ( 1, 1), Σ = {v} ( 1, 1) A naive expectation for the space of cone hyp str H on (M, Σ, θ) (0 θ < 2π) is as follows: 6

10 What is the space H? As the first step, we consider a subspace H H consisting of hyp str with some finiteness condition. There are 3 candidates for the reasonable condition: For a cone hyperbolic structure σ on (M, Σ, θ), 1. σ is convex cocompact (Moroianu-Schlenker). 2. The holonomy representation of σ satisfies the BQ-condition (Bowditch, Tan-Wong-Zhang). 3. σ is induced from a good fundamental polyhedron, defined as a variation of Jorgensen s theory. 7

11 What is the space H? As the first step, we consider a subspace H H consisting of hyp str with some finiteness condition. There are 3 candidates for the reasonable condition: For a cone hyperbolic structure σ on (M, Σ, θ), 1. σ is convex cocompact (Moroianu-Schlenker). 2. The holonomy representation of σ satisfies the BQ-condition (Bowditch, Tan-Wong-Zhang). 3. σ is induced from a good fundamental polyhedron, defined as a variation of Jorgensen s theory. Conjecture (A-Yamashita)

12 The real slice H R H H R = {σ H ρ σ is conjugate to a P SL(2, R)-repr} where ρ σ : π 1 (T 0 ) P SL(2, C): the hol repr for σ. Theorem (Cor. to Goldman, Tan-Wong-Zhang) The conditions 1 and 2 are equavalent in H R. Theorem (A) The conditions 2 and 3 are equavalent in H R. 8

13 Parametrization for H There are 2 invariants: Conformal str of the Riemann surface at infinity for convex cocompact hyp str. This restricts to a parameter for QF (Ahlfors-Bers theory). The side parameter for a hyp str with a good fund polyhedron. This restricts to a parameter for QF (Jorgensen theory). 9

14 Parametrization for H There are 2 invariants: Conformal str of the Riemann surface at infinity for convex cocompact hyp str. This restricts to a parameter for QF (Ahlfors-Bers theory). The side parameter for a hyp str with a good fund polyhedron. This restricts to a parameter for QF (Jorgensen theory). Main Theorem The side parameter restricts to a parameter for H R. 9

15 Good fundamental polyhedron for H R The good fund polyh for H R is combinatorially equiv to 10

16 Comparison with the classical case θ = 0: classical case 0 < θ < 2π: our case 11

17 Comparison with the classical case θ = 0: classical case 0 < θ < 2π: our case Prop A good fund polyh is the Ford domain, in the sense that it is the completion of the complement of the cut locus with respect to an embedded standard horoball with cone singularity. 11

18 Elliptic generators By the covering π 1 (T 0 ) π 1 (O 0 ) π 1 (T 0 ) = A, B where π 1 (O 0 ) = P, Q, R P 2, Q 2, R 2 K = RQP A = KP, B = K 1 R P, Q, R is an elliptic generator triple {P j } =..., K 1 R(K 1 ) 1, P, Q, R, KP K 1, KQK 1, KRK 1, K 2 P K 2,... a sequence of elliptic generator 12

19 Elliptic generators and Farey tessellation Fact Each seq of ellip gen corresponds to a Farey triangle. Two adjacent seq of ellip gen are related by (P, Q, R) (P, R, RQR). 13

20 Side parameter 1 Fact 1. The faces of good fund polyh correspond to the seq of elliptic generators {P j } associated with a triangle in D. 14

21 Side parameter 2 At each face corresponding to P j, let = (angle of the visible part)/2. θ ± j Fact 2. For each ϵ {, +}, θ ϵ 0 + θϵ 1 + θϵ 2 = (2π ϵθ)/4. 15

22 Side parameter 3 For each ϵ {, +}, the side parameter map ν ϵ : Hθ R H2 is defined as follows, where Hθ R = {cone angle = θ} HR : For σ Hθ R, ν ϵ (σ) is in the triangle determined by Fact 1. In, ν ϵ (σ) is determined by the barycentric coord. θ ϵ 0 + θϵ 1 + θϵ 2 = (2π ϵθ)/4. 16

23 Main Theorem (again) Main Theorem For each ϵ {, +} and θ (0, 2π), the map ν ϵ : Hθ R H2 is a homeomorphism. 17

24 Main Theorem (again) Main Theorem For each ϵ {, +} and θ (0, 2π), the map ν ϵ : H R θ H2 is a homeomorphism. Idea of Proof. Fix 0 < θ < 2π, ϵ {, +} and a seq of ell gen {P j }, and pick θ j s.t. θ 0 0, θ 1 > 0, θ 2 > 0 and θ 0 + θ 1 + θ 2 = (2π ϵθ)/4. We show that (θ 0, θ 1, θ 2 ) is uniquely realizable as the side parameter for a good fund polyh whose faces correspond to P j (j Z). By looking at the polyhedron carefully, the proof is reduced to the unique existence of elementary pieces : For example, when ϵ = +, find triangles for j {0, 1, 2} such that: l > 0 is independent of j. φ 0 + φ 1 + φ 2 = θ/4. 17