Plumbing Constructions in Quasifuchsian Space.

Transcription

1 Plumbing Constructions in Quasifuchsian Space. Sara Maloni University of Warwick, University of Toulouse August 8, 2012

2 Table of content 1 Dehn Thurston coordinates 2 Maskit embedding and pleating ray 3 Gluing construction 4 Main theorems 5 Other slices of Quasifuchsian Space

3 Dehn Thurston coordinates Given a pants decomposition P = {σ 1,..., σ ξ } on a surface Σ, Dehn defined an injection i : S = S(Σ) Z ξ 0 Zξ by i(γ) = (q 1 (γ),..., q ξ (γ); tw 1 (γ),..., tw ξ (γ)). 1 q i (γ) = i(γ, σ i ) Z 0 are the length parameters; 2 tw i (γ) Z are the twist parameters of γ. A B A B = E C D C D Figure: Penner and Harer twist ˆp i = 1 and D. Thurston s twist p i = 0.

4 Relation between ˆp i and p i Suppose two pairs of pants meet along σ = E PC. Label their respective boundary curves (A, B, E) and (C, D, E) in clockwise order. Theorem (M Series) Let γ S and let ˆp i and p i denote the PH twist and the DT twist around σ. Then ˆp i = p i + l(a, E; B) + l(c, E; D) q i, 2 where l(x, Y ; Z) denotes the number of strands of γ P running from the boundary curve X to the boundary curve Y in the pair of pants P = (X, Y, Z).

5 Thurston s symplectic form Let τ Th be Thurston symplectic form on S ML Q (Σ). Theorem (M.) Suppose that loops γ, γ S belongs to the same chart and let i(γ) = (q 1,..., q ξ ; p 1,..., p ξ ), i(γ ) = (q 1,..., q ξ ; p 1,..., p ξ ) their DT coordinates. Then τ Th (γ, γ ) = 1 2 ξ (q i p i q ip i ). i=1 In addition, if γ, γ are disjoint, then τ Th (γ, γ ) = 0.

6 Basic definitions on Kleinian groups PSL(2, C) acts on H 3 by isometries and on Ĉ = C by conformal maps. Definition A Kleinian group G is a discrete (torsion-free) subgroup of PSL(2, C). The limit set Λ(G) is the set of accumulation points of the action of G on Ĉ. The regular set Ω(G) is Ĉ Λ(G). A Fuchsian group is a discrete subgroup of PSL(2, R), or, equivalently, a Kleinian group G such that Λ(G) is a circle. A Quasifuchsian group is a Kleinian group G such that Λ(G) is a topological circle, or, equivalently, a quasi-conformal deformation of a Fuchsian group.

7 The Maskit embedding The Maskit slice M is the set of representations ρ : π 1 (Σ) PSL(2, C) (up to conjugation in PSL(2, C)) such that: 1 G ρ = ρ (π 1 (Σ)) is discrete and ρ is an isomorphism, 2 ρ(σ i ) are parabolic, 3 all components of Ω(G) are simply connected and there is exactly one invariant component Ω + (G), 4 Ω + (G)/G is homeomorphic to Σ and the other components are triply punctured spheres. Figure: Quasifuchsian Group and Maskit Group.

8 DT coordinates Maskit embedding Gluing construction Main theorems Other slices Picture of the Maskit embedding for the once punctured torus Σ1,1 Figure: The Maskit embedding M(Σ1,1 ) for the once punctured torus. Picture courtesy David Wright.

9 Pleating Ray A pleated surface is a a hyperbolic surface which is totally geodesic almost everywhere and such that the locus of points where it fails to be totally geodesic is a geodesic lamination. By Thurston, each component of the boundary C(G)/G of the convex core is a pleated surface. Given ρ M, denote β(ρ) ML(Σ) the bending lamination of C + /G ρ, where G ρ = ρ (π 1 (Σ)). Given [η] PML(Σ), the pleating ray P = P [η] of [η] is the set of elements ρ M for which β(ρ) [η].

10 Gluing construction Let Σ be a surface with χ(σ) < 0 and let PC = {σ 1,..., σ ξ } be a pants decomposition on it. Let µ = (µ 1,..., µ ξ ) H ξ STEP 1: Any triply punctured sphere is isometric to P = H/Γ, where ( ) ( ) Γ =, Identify any P i to the fundamental domain of Γ by the homeomorphisms Φ i : int(p i ).

11 Gluing (P 0(P 0 ) STEP 2: Let σ i = ɛ P ɛ P, then the gluing is described by P ˆ ˆ 0 P 0 z z 0 Ω 1 ɛ J 1 Tµ 1 i Ω ɛ H1 H 0 0 where µ i H is the gluing parameter ( and ) Ω = Id, ( ) Ω 0 =, Ω =, 1 1 ( ) ( ) i 0 1 µi J =, T 0 i µi = J Tµ 0 1(z) 0(z 0 ) J 0(z)

12 Projective structure This describes a (complex) projective structure on Σ, which depends on the gluing parameter µ = (µ 1,..., µ ξ ) H ξ. In particular, given µ H ξ, we define a developing map Dev µ : Σ Ĉ and a holonomy map ρ µ : π 1 (Σ) PSL(2, C). Theorem (M Series) If Dev µ : Σ Ĉ is an embedding, then ρµ is a group isomorphism and ρ µ M. In addition, these representations ρ µ parametrise M.

13 Top Terms Formula Let ρ µ : π 1 (Σ) PSL(2, C) be the holonomy described by the gluing construction. Let γ be a simple closed curve on Σ, not parallel to any of the pants curves σ i. Theorem (Top Terms Formula, M Series) ( Tr ρ µ (γ) = ±i q 2 h µ 1 + (p ) q1 ( 1 q 1 ) µ ξ + (p ) qξ ξ q ξ ) + R, q 1 q ξ where q = ξ i=1 q i > 0; R represents terms with total degree in µ 1 µ ξ at most q 2; h = h(γ) is the total number of waves.

14 Asymptotic direction of pleating rays Theorem (Asymptotic direction, M, Series, Keen Series) Suppose that η = ξ i=1 a iγ i is an admissible measured lamination on Σ. Then, as the bending measure β(g µ ) [η] tends to zero, the pleating ray P [η] in M approaches the line Rµ i = p i(η) q i (η) + 1, Iµ 1 Iµ j = q j(η) q 1 (η), where (q 1 (η),..., q ξ (η); p 1 (η),..., p ξ (η)) are the Dehn Thurston coordinates for η.

15 Generalised gluing construction Given a pants decomposition PC = {σ 1,..., σ ξ } on Σ, let c = (c 1,..., c ξ ) R ξ + and µ = (µ 1,..., µ ξ ) (C/2iπ) ξ. We describe a (complex) projective structure on Σ with developing map Dev c,µ and holonomy map ρ c,µ. In particular, ρ c,µ (γ) is hyperbolic and Tr ρ c,µ (γ) = ±2 cosh(c j ). Theorem (M.) If c 0 keeping µ fixed, where µ i = iπ µ i c i, then ρ c,µ ρ µ.

16 Linear slices L c Given PC = {σ 1,..., σ ξ }, the complex Fenchel Nielsen coordinates FN C : QF(Σ) (C + /2iπ) ξ (C/2iπ) ξ are defined by FN C (G) = (λ 1,..., λ ξ, τ 1,..., τ ξ ), where λ i are the complex length and τ i are the complex twist of the pants curve σ i. Definition Given c R ξ +, we define the c slice (or the linear slice) L c to be the set L c = {(c, τ) FN C (QF(Σ)) sign(iτ 1 ) =... = sign(iτ ξ )}.

17 Connectedness of linear slices L c Figure: The linear slice L c when c = 1, 2, 4, 5, 10, 20.

18 Connectedness of linear slices L c

19 End