Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds 吉田 建一 (Ken’ichi YOSHIDA) Department of Mathematics, Kyoto University October 20, 2017 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 1 / 22
The hyperbolic space t H 3 : the hyperbolic 3-space ∼ y = { ( x , y , t ) ∈ R 3 | t > 0 } with the metric ds 2 = ( dx 2 + dy 2 + dt 2 ) / t 2 x (the upper half-space model) geodesic in H 3 — circular arc or line orthogonal to the plane { ( x , y , 0) ∈ R 3 } Identifications: { ( x , y , 0) ∈ R 3 } ∼ = C ( ∋ x + iy ) ∂ H 3 ∼ = C ∪ {∞} ori.-preserving isometry of H 3 ← → M¨ obius transformation of C ∪ {∞} Isom + ( H 3 ) ∼ = PSL (2 , C ) (= SL (2 , C ) / ± 1) [ a ] = { z = x + iy �→ az + b b ∈ PSL (2 , C ) ∼ cz + d } c d 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 2 / 22
Types of isometries of H 3 An orientation-preserving isometry of H 3 is one of the following types: identity elliptic — fixing pointwise a geodesic in H 3 parabolic — fixing a single point in ∂ H 3 hyperbolic (loxodromic) — fixing two points in ∂ H 3 (fixing setwise a geodesic in H 3 ) parabolic hyperbolic elliptic 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 3 / 22
Types and traces The types are determined by the trace. [ λ ] 0 elliptic ∼ ∈ PSL (2 , C ) ( | λ | = 1) ⇐ ⇒ − 2 < trace < 2 λ − 1 0 [ 1 ] 1 parabolic ∼ ∈ PSL (2 , C ) ⇐ ⇒ trace = ± 2 0 1 [ λ ] 0 hyperbolic ∼ ∈ PSL (2 , C ) ( | λ | ̸ = 1) ⇐ ⇒ trace / ∈ [ − 2 , 2] λ − 1 0 (up to conjugacy) 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 4 / 22
Hyperbolic 3-manifold M : an orientable hyperbolic 3-manifold (hyperbolic : ⇐ ⇒ having a complete metric of sectional curvature − 1) ⇒ M ∼ = H 3 /π 1 ( M ), where π 1 ( M ) is regarded as a discrete subgroup of = PSL (2 , C ). elliptic element / ∈ π 1 ( M ) parabolic element ∈ π 1 ( M ) ← → loop in a “cusp” of M hyperbolic element ∈ π 1 ( M ) ← → closed geodesic in M (up to conjugacy) 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 5 / 22
Cusp of a hyperbolic 3-manifold If vol ( M ) < ∞ , then M is homeomorphic to the interior of a compact 3-manifold M with boundary consisting tori. In this case, a cusp of M is a component of ∂ M . (In general, there may be annular cusps in the boundary.) A neighborhood of a torus cusp is isometric to { ( x , y , t ) ∈ H 3 | t > t 0 } / ⟨ z �→ z + 1 , z �→ z + τ ⟩ for some t 0 > 0 and τ ∈ C with Im ( τ ) > 0. ( z = x + iy .) τ is called the modulus of the cusp T with respect to fixed generators of π 1 ( T ). There are many links whose complement is hyperbolic. For example: 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 6 / 22
3-punctured sphere in a hyperbolic 3-manifold A 3-punctured sphere (a.k.a. a pair of pants) is an orientable surface of genus 0 with 3 punctures. A totally geodesic 3-punctured sphere is the double of an ideal hyperbolic triangle. The (complete) hyperbolic structure of a 3-punctured sphere is unique. Theorem (Adams (1985)) An essential (properly embedded) 3-punctured sphere in a hyperbolic 3-manifold is isotopic to a totally geodesic 3-punctured sphere. By taking conjugacies, [ 1 ] [ 1 ] 2 0 we may assume x = and y = . 0 1 c 1 y x tr ( xy − 1 ) = ± 2 ⇐ ⇒ c = 0 or 2. Then we have c = 2. 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 7 / 22
Union of 3-punctured spheres A n ( n ≥ 1) B 2 n ( n ≥ 1) T 3 T 4 ...etc. The index indicates the number of 3-punctured spheres. 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 8 / 22
Cusp modulus for A n The metric of neighborhood of 3-punctured spheres of the type A n for n ≥ 2 is determined by the modulus τ ∈ C of cusps. (The moduli of the adjacent cusps coincide.) y z w x C 2 C 1 Σ 1 Σ 2 We may assume that [ 1 [ 1 ] [ 1 ] [ 1 ] ] 2 0 2 /τ 0 x = , y = , z = , w = ∈ PSL (2 , C ). 0 1 2 1 0 1 2 τ 1 [ − 3 [ − 3 ] ] 2 2 /τ ( xy − 1 = and zw − 1 = are parabolic.) − 2 1 − 2 τ 1 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 9 / 22
Bounds of moduli Consider the set C n := { τ ∈ C | τ is the modulus for 3-punctured spheres of A n contained in a hyperbolic 3-manifold } . (not assumed to have finite volume) Lemma (The Shimizu-Leutbecher lemma) Suppose that a group generated by two elements [ 1 ] [ a ] 2 b , ∈ PSL (2 , C ) is discrete. Then c = 0 or | c | ≥ 1 / 2 . 0 1 c d We apply the Shimizu-Leutbecher lemma for [ 1 ] [ ] 2 1 0 , y n w m = ⟨ x = ⟩ and 0 1 2 m τ + 2 n 1 [ 1 ] [ 1 ] 0 (2 m /τ ) + 2 n , x n z m = ⟨ y = ⟩ . 2 1 0 1 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 10 / 22
Rough bound Proposition τ ∈ C 2 satisfies the following inequalities: | m τ + n | ≥ 1 / 4 and | ( m /τ ) + n | ≥ 1 / 4 for ( m , n ) ∈ Z × Z \ (0 , 0) . In particular, 1 4 ≤ | τ | ≤ 4 and 0 . 079 ≤ arg τ ≤ π − 0 . 079 . 4 i i 1 4 i − 4 0 4 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 11 / 22
Sets of moduli Consider the 4-punctured sphere Σ near the 3-punctured spheres of A n in a hyperbolic 3-manifold M . (Σ bounds M τ n .) Σ M τ n C incomp := { τ ∈ C n | Σ is incompressible } n C comp := { τ ∈ C n | Σ is compressible } n C n = C incomp ∪ C comp n n 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 12 / 22
The case that Σ is incompressible τ ∈ C incomp = ⇒ M τ n extends an infinite volume hyperbolic 3-manifold n homeomorphic to M τ n . ) ∼ int ( C incomp = the Teichm¨ uller space of Σ (homeomorphic to an open n disk) C incomp = cl ( int ( C incomp )) (homeomorphic to a closed disk) n n ∂ C incomp ∼ = R ∪ {∞} n rational point in R or ∞ ← → cusp irrational point in R ← → ending lamination Σ Example: slope = 3 / 2 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 13 / 22
The case that Σ is compressible τ ∈ C comp = ⇒ M = M τ n ∪ (a trivial tangle). n C comp ← → Q \ { 0 , 1 , 2 } 2 C comp ← → Q \ { 0 } 3 C comp ← → Q ( n ≥ 4) n 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 14 / 22
Shape of C n C incomp C comp n n 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 15 / 22
For hyperbolic 3-manifolds of finite volume C fin := { τ ∈ C | τ is the modulus for 3-punctured spheres of A n contained n in a hyperbolic 3-manifold of finite volume } C comp ⊂ C fin n . n Theorem (Brooks (1986)) Let Γ < PSL (2 , C ) be a geometrically finite Kleinian group. Then there exist arbitrarily small quasi-conformal deformations Γ ϵ of Γ , such that Γ ϵ admits an extension of the fundamental group of a finite volume hyperbolic 3-manifold. Corollary C fin is dense in C n . n 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 16 / 22
Way to compute To plot points of ∂ C incomp , consider the condition that a simple loop in Σ n is an annular cusp. Solve equations: − 2 = trace of an element represented by a simple loop in Σ. We will not avoid plotting unnecessary points outside C incomp . n Remark: C n +1 ⊂ C n , C incomp ⊂ C incomp , C n +2 ⊂ C incomp . n +1 n n 吉田 建一 (Ken’ichi YOSHIDA) (Department of Mathematics, Kyoto University) Parametrization for intersecting 3-punctured spheres in hyperbolic 3-manifolds October 20, 2017 17 / 22
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