Geometric structures on the Figure Eight Structures Martin Deraux - PowerPoint PPT Presentation

Geometric structures on the Figure Eight Knot Complement ICERM Workshop Exotic Geometric Geometric structures on the Figure Eight Structures Martin Deraux Knot Complement The figure eight knot Presentation Holonomy Prism picture ICERM Workshop Tetrahedron picture Spherical CR Exotic Geometric Structures geometry Limit set Complex hyperbolic geometry Central question Known results Martin Deraux Main theorem Horotube Combinatorics Institut Fourier - Grenoble Triangle group Surgery Rank 1 boundary unipotent Sep 16, 2013

Geometric The figure eight knot structures on the Figure Eight Knot Complement Various pictures of 4 1 : ICERM Workshop Exotic Geometric Structures Martin Deraux The figure eight knot Presentation Holonomy Prism picture Tetrahedron picture Spherical CR geometry Limit set Complex hyperbolic geometry Central question Known results Main theorem Horotube Combinatorics Triangle group Surgery Rank 1 boundary unipotent K = figure eight knot

Geometric The complete (real) hyperbolic structure structures on the Figure Eight Knot Complement M = S 3 \ K carries a complete hyperbolic metric ICERM Workshop Exotic Geometric Structures M can be realized as a quotient Martin Deraux Γ \ H 3 The figure eight R knot Presentation where Γ ⊂ PSL 2 ( C ) is a lattice (discrete group with quotient Holonomy Prism picture Tetrahedron picture of finite volume) Spherical CR geometry ◮ One cusp with cross-section a torus. Limit set Complex hyperbolic ◮ Discovered by R. Riley (1974) geometry Central question Known results ◮ Part of a much more general statement about knot Main theorem Horotube complements/3-manifolds (Thurston) Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Geometric Holonomy representation structures on the Figure Eight Knot Complement For example by Wirtinger, get ICERM Workshop Exotic Geometric π 1 ( M ) = � g 1 , g 2 , g 3 | g 1 g 2 = g 2 g 3 , g 2 = [ g 3 , g − 1 Structures 1 ] � , Martin Deraux with fundamental group of the boundary torus generated by The figure eight knot Presentation g 1 and [ g − 1 3 , g 1 ][ g − 1 1 , g 3 ] Holonomy Prism picture Tetrahedron picture Alternatively Spherical CR geometry Limit set π 1 ( M ) = � a , b , t | tat − 1 = aba , tbt − 1 = ab � . Complex hyperbolic geometry Central question Known results The figure eight knot complement fibers over the circle, with Main theorem Horotube Combinatorics punctured torus fiber. Triangle group Surgery Rank 1 boundary unipotent

Geometric Holonomy representation (cont.) structures on the Figure Eight Knot Complement ICERM Workshop Exotic Geometric Search for ρ : π 1 ( M ) → PSL 2 ( C ) with ρ ( g 1 ) = G 1 , Structures ρ ( g 3 ) = G 3 , Martin Deraux � 1 � � 1 � The figure eight 1 0 knot G 1 = , G 3 = Presentation 0 1 − a 1 Holonomy Prism picture Tetrahedron picture Requiring Spherical CR G 1 [ G 3 , G − 1 1 ] = [ G 3 , G − 1 1 ] G 3 geometry Limit set Complex hyperbolic get a 2 + a + 1 = 0, so geometry Central question Known results √ Main theorem a = − 1 ± i 3 Horotube = ω or ω. Combinatorics 2 Triangle group Surgery Rank 1 boundary unipotent

Geometric Ford domain for the image of ρ structures on the Figure Eight Knot Complement √ Bounded by unit spheres centered in Z [ ω ], ω = − 1+ i 3 2 ICERM Workshop √ Exotic Geometric Cusp group generated by translations by 1 and 2 i 3. Structures Martin Deraux The figure eight knot Presentation Holonomy Prism picture Tetrahedron picture Spherical CR geometry Limit set Complex hyperbolic geometry Central question Known results Main theorem Horotube Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Geometric Prism picture structures on the Figure Eight Knot Complement ICERM Workshop Exotic Geometric Structures x 3 x 1 x 1 x 4 x 3 x 3 Martin Deraux x 2 x 4 x 4 x 2 The figure eight knot x 2 Presentation x 1 x 1 x 1 x 4 x 3 x 3 Holonomy Prism picture x 4 x 4 Tetrahedron picture Spherical CR geometry x 3 x 2 Limit set Complex hyperbolic geometry Central question Known results Main theorem Horotube Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Geometric Triangulation picture structures on the Figure Eight Knot Can also get the hyperbolic structure gluing two ideal Complement tetrahedra , with invariants z , w . Compatibility equations: ICERM Workshop Exotic Geometric Structures z ( z − 1) w ( w − 1) = 1 Martin Deraux For a complete structure, ask the boundary holonomy to The figure eight knot have derivative 1, and this gives Presentation Holonomy Prism picture z = w = ω Tetrahedron picture Spherical CR geometry Limit set Complex hyperbolic geometry Central question Known results Main theorem Horotube Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Geometric Complete spherical CR structures structures on the Figure Eight Knot Complement ICERM Workshop Exotic Geometric Structures Spherical CR structure arising as the boundary of a ball Martin Deraux quotient . The figure eight knot Ball quotient: Γ \ B 2 , where Γ is a discrete subgroup of Presentation Holonomy Bihol ( B 2 ) = PU (2 , 1). Prism picture Tetrahedron picture Spherical CR geometry The manifold at infinity inherits a natural spherical CR Limit set structure, called “complete” or “uniformizable”. Complex hyperbolic geometry Central question Known results Main theorem Horotube Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Geometric Domain of discontinuity structures on the Figure Eight Knot Complement ICERM Workshop Γ ⊂ PU (2 , 1) discrete Exotic Geometric Structures ◮ Domain of discontinuity Ω Γ Martin Deraux ◮ Limit set Λ Γ = S 3 − Ω Γ The figure eight knot Presentation The orbifold/manifold at infinity of Γ is Γ \ Ω Γ Holonomy Prism picture Tetrahedron picture ◮ Manifold only if no fixed points in Ω Γ (isolated fixed Spherical CR points inside B 2 are OK); geometry Limit set Complex hyperbolic ◮ Can be empty (e.g. when Γ is (non-elementary) and a geometry Central question normal subgroup in a lattice). Known results Main theorem Horotube Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Biholomorphisms of B 2 Geometric structures on the Figure Eight Knot Complement Up to scaling, B 2 carries a unique metric invariant under the PU (2 , 1)-action, the Bergman metric. ICERM Workshop Exotic Geometric B 2 ⊂ C 2 ⊂ P 2 Structures C Martin Deraux With this metric: complex hyperbolic plane . The figure eight ◮ Biholomorphisms of B 2 : restrictions of projective knot Presentation transformations (i.e. linear tsf of C 3 ). Holonomy Prism picture ◮ A ∈ GL 3 ( C ) preserves B 2 if and only if Tetrahedron picture Spherical CR geometry Limit set A ∗ HA = H Complex hyperbolic geometry Central question Known results where   Main theorem − 1 0 0 Horotube Combinatorics   H = 0 1 0 Triangle group Surgery 0 0 1 Rank 1 boundary unipotent

Geometric structures on the Figure Eight Knot Complement Equivalent Hermitian form:   0 0 1 ICERM Workshop   Exotic Geometric J = 0 1 0 Structures 1 0 0 Martin Deraux The figure eight Siegel half space: knot Presentation 2 Im ( w 1 ) + | w 2 | 2 < 0 Holonomy Prism picture Tetrahedron picture Spherical CR Boundary at infinity ∂ ∞ H 2 C (minus a point) should be geometry Limit set viewed as the Heisenberg group , C × R with group law Complex hyperbolic geometry Central question Known results ( z , t ) ∗ ( w , s ) = ( z + w , t + s + 2 Im ( z ¯ w )) . Main theorem Horotube Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Geometric structures on the Figure Eight Knot Complement ◮ Copies of H 1 C (affine planes in C 2 ) have curvature − 1 R ( R 2 ⊂ C 2 ) have curvature − 1 / 4 (linear ◮ Copies of H 2 ICERM Workshop only when through the origin) Exotic Geometric Structures ◮ No totally geodesic embedding of H 3 R ! Martin Deraux The figure eight knot For this normalization, we have Presentation Holonomy Prism picture cosh 1 |� Z , W �| Tetrahedron picture 2 d ( z , w ) = � Spherical CR � Z , Z �� W , W � geometry Limit set Complex hyperbolic where geometry Central question Known results ◮ Z are homogeneous coordinates for z Main theorem Horotube ◮ W are homogeneous coordinates for w Combinatorics Triangle group Surgery Rank 1 boundary unipotent

Isometries of H 2 Geometric structures on the C Figure Eight Knot Complement Classification of (non-trivial) isometries ◮ Elliptic ( ∃ fixed point inside) ICERM Workshop Exotic Geometric ◮ regular elliptic (three distinct eigenvalues) Structures ◮ complex reflections Martin Deraux ◮ in lines The figure eight ◮ in points knot Presentation ◮ Parabolic (precisely one fixed point in ∂ H 2 C ) Holonomy Prism picture ◮ Unipotent (some representative has 1 as its only Tetrahedron picture Spherical CR eigenvalue) geometry ◮ Screw parabolic Limit set Complex hyperbolic ◮ Loxodromic (precisely two fixed points in ∂ H 2 geometry C ) Central question Known results Main theorem Horotube Combinatorics PU (2 , 1) has index 2 in Isom H 2 C (complex conjugation). Triangle group Surgery Rank 1 boundary unipotent