Non-geometric veering triangulations Ahmad Issa University of Texas at Austin Joint work with Craig Hodgson and Henry Segerman 1
◮ Today we consider ideal triangulations. An ideal triangulation of a 3-manifold M is a decomposition of M into tetrahedra with their vertices removed. 2
◮ Today we consider ideal triangulations. An ideal triangulation of a 3-manifold M is a decomposition of M into tetrahedra with their vertices removed. ◮ In 2010 Ian Agol gave an elegant construction of ideal triangulations of a subclass of hyperbolic 3-manifolds, characterised by a combinatorial condition which he called veering . 2
◮ Today we consider ideal triangulations. An ideal triangulation of a 3-manifold M is a decomposition of M into tetrahedra with their vertices removed. ◮ In 2010 Ian Agol gave an elegant construction of ideal triangulations of a subclass of hyperbolic 3-manifolds, characterised by a combinatorial condition which he called veering . ◮ He posed the question: is every veering triangulation geometric , i.e. realised by positive volume ideal hyperbolic tetrahedra in the complete hyperbolic structure? 2
◮ Today we consider ideal triangulations. An ideal triangulation of a 3-manifold M is a decomposition of M into tetrahedra with their vertices removed. ◮ In 2010 Ian Agol gave an elegant construction of ideal triangulations of a subclass of hyperbolic 3-manifolds, characterised by a combinatorial condition which he called veering . ◮ He posed the question: is every veering triangulation geometric , i.e. realised by positive volume ideal hyperbolic tetrahedra in the complete hyperbolic structure? ◮ Previously, this was verified for many examples. 2
◮ Today we consider ideal triangulations. An ideal triangulation of a 3-manifold M is a decomposition of M into tetrahedra with their vertices removed. ◮ In 2010 Ian Agol gave an elegant construction of ideal triangulations of a subclass of hyperbolic 3-manifolds, characterised by a combinatorial condition which he called veering . ◮ He posed the question: is every veering triangulation geometric , i.e. realised by positive volume ideal hyperbolic tetrahedra in the complete hyperbolic structure? ◮ Previously, this was verified for many examples. ◮ Answer: No! We found examples of non-geometric veering triangulations. 2
◮ Today we consider ideal triangulations. An ideal triangulation of a 3-manifold M is a decomposition of M into tetrahedra with their vertices removed. ◮ In 2010 Ian Agol gave an elegant construction of ideal triangulations of a subclass of hyperbolic 3-manifolds, characterised by a combinatorial condition which he called veering . ◮ He posed the question: is every veering triangulation geometric , i.e. realised by positive volume ideal hyperbolic tetrahedra in the complete hyperbolic structure? ◮ Previously, this was verified for many examples. ◮ Answer: No! We found examples of non-geometric veering triangulations. ◮ Conjecture: Every cusped hyperbolic 3-manifold admits a geometric ideal triangulation. 2
Mapping torus of a homeomorphism 3
Mapping torus of a homeomorphism ◮ The veering triangulations we are interested in are triangulations of a subclass of mapping tori. 3
Mapping torus of a homeomorphism ◮ The veering triangulations we are interested in are triangulations of a subclass of mapping tori. ◮ The mapping torus of a homeomorphism ϕ : S → S of a surface S is the 3-manifold M ϕ = S × [0 , 1] / { ( x , 0) ∼ ( ϕ ( x ) , 1) } . S x {1} φ S x {0} 3
Mapping torus of a homeomorphism ◮ The veering triangulations we are interested in are triangulations of a subclass of mapping tori. ◮ The mapping torus of a homeomorphism ϕ : S → S of a surface S is the 3-manifold M ϕ = S × [0 , 1] / { ( x , 0) ∼ ( ϕ ( x ) , 1) } . S x {1} φ S x {0} Hyperbolization theorem for mapping tori The mapping torus M ϕ admits a hyperbolic structure if and only if ϕ is (isotopic to) a pseudo-Anosov homeomorphism. 3
1 1 ) : R 2 → R 2 . ◮ Consider A = ( 2 1 A 4
1 1 ) : R 2 → R 2 . ◮ Consider A = ( 2 1 ◮ Descends to a homeomorphism ϕ : S → S where S = ( R 2 − Z 2 ) / Z 2 is the once-punctured torus. A 4
1 1 ) : R 2 → R 2 . ◮ Consider A = ( 2 1 ◮ Descends to a homeomorphism ϕ : S → S where S = ( R 2 − Z 2 ) / Z 2 is the once-punctured torus. ◮ A has eigenvectors u and v with eigenvalues λ > 1 and 1 λ < 1. v 4 4 contract u 2 2 A stretch 0 0 � 2 � 2 � 4 � 4 4 � 4 � 2 0 2 4 � 4 � 2 0 2 4
1 1 ) : R 2 → R 2 . ◮ Consider A = ( 2 1 ◮ Descends to a homeomorphism ϕ : S → S where S = ( R 2 − Z 2 ) / Z 2 is the once-punctured torus. ◮ A has eigenvectors u and v with eigenvalues λ > 1 and 1 λ < 1. ◮ Lines parallel to u foliate (partition) R 2 − Z 2 . Projecting these lines onto S gives a foliation F u of S into leaves (curves). v 4 4 contract u 2 2 A stretch 0 0 � 2 � 2 � 4 � 4 4 � 4 � 2 0 2 4 � 4 � 2 0 2 4
1 1 ) : R 2 → R 2 . ◮ Consider A = ( 2 1 ◮ Descends to a homeomorphism ϕ : S → S where S = ( R 2 − Z 2 ) / Z 2 is the once-punctured torus. ◮ A has eigenvectors u and v with eigenvalues λ > 1 and 1 λ < 1. ◮ Lines parallel to u foliate (partition) R 2 − Z 2 . Projecting these lines onto S gives a foliation F u of S into leaves (curves). ◮ Measure µ on F assigns a positive real number to an arc transverse to F on S . v 4 4 contract u 2 2 A stretch 0 0 � 2 � 2 � 4 � 4 4 � 4 � 2 0 2 4 � 4 � 2 0 2 4
1 1 ) : R 2 → R 2 . ◮ Consider A = ( 2 1 ◮ Descends to a homeomorphism ϕ : S → S where S = ( R 2 − Z 2 ) / Z 2 is the once-punctured torus. ◮ A has eigenvectors u and v with eigenvalues λ > 1 and 1 λ < 1. ◮ Lines parallel to u foliate (partition) R 2 − Z 2 . Projecting these lines onto S gives a foliation F u of S into leaves (curves). ◮ Measure µ on F assigns a positive real number to an arc transverse to F on S . ◮ ϕ ( F u ) = F u and ϕ ( µ u ) = λµ u . v 4 4 contract u 2 2 A stretch 0 0 � 2 � 2 � 4 � 4 4 � 4 � 2 0 2 4 � 4 � 2 0 2 4
1 1 ) : R 2 → R 2 . ◮ Consider A = ( 2 1 ◮ Descends to a homeomorphism ϕ : S → S where S = ( R 2 − Z 2 ) / Z 2 is the once-punctured torus. ◮ A has eigenvectors u and v with eigenvalues λ > 1 and 1 λ < 1. ◮ Lines parallel to u foliate (partition) R 2 − Z 2 . Projecting these lines onto S gives a foliation F u of S into leaves (curves). ◮ Measure µ on F assigns a positive real number to an arc transverse to F on S . ◮ ϕ ( F u ) = F u and ϕ ( µ u ) = λµ u . ◮ Starting with v instead of u gives another measured foliation ( F s , µ s ). v 4 4 contract u 2 2 A stretch 0 0 � 2 � 2 � 4 � 4 4 � 4 � 2 0 2 4 � 4 � 2 0 2 4
◮ A singular foliation F of a surface S is a foliation of S away from a finite set P of singular points . Points of P are locally modelled on a k -prong with k ≥ 3. Punctures of S are locally modelled on a k -prong with k ≥ 1. Figure : k -prong for k = 1 and k = 3, singular point (or puncture) shown in red. 5
A homeomorphism ϕ : S → S (with ( S ) < 0), is pseudo-Anosov if there exist measured foliations ( F u , µ u ), ( F s , µ s ) of S with transverse leaves and a common set of singular points with ϕ ( F u , µ u ) = ( F u , λµ u ) and ϕ ( F s , µ s ) = ( F s , 1 λ µ s ), where λ > 1 is called the dilatation of ϕ . ◮ ( F u , µ u ) is called the unstable foliation of ϕ and ( F s , µ s ) is called the stable foliation. ◮ A finite segment of a leaf of F u can be thought of as stretched by a factor of λ . 6
Train tracks ◮ A (measured) train track on a surface S is a finite 1-complex τ ⊂ S with C 1 embedded edges and a positive real number assigned to each edge such that each vertex is modelled on a switch (see Figure). a, b, c > 0 b Switch condition: a = b + c a c 1 2.618.. 1.618.. 7
Train tracks ◮ A (measured) train track on a surface S is a finite 1-complex τ ⊂ S with C 1 embedded edges and a positive real number assigned to each edge such that each vertex is modelled on a switch (see Figure). a, b, c > 0 b Switch condition: a = b + c a c ◮ We require that each component of S \ τ is a disk with at least 3 cusps, or a punctured disk with at least 1 cusp. 1 2.618.. 1.618.. 7
Train track to foliation ◮ Step 1: Replace each edge of τ by a foliated Euclidean rectangle. 1.618... 1.618... anything 8
Train track to foliation ◮ Step 1: Replace each edge of τ by a foliated Euclidean rectangle. 1.618... 1.618... anything 1 2.618... 1.618... 8
Train track to foliation ◮ Step 1: Replace each edge of τ by a foliated Euclidean rectangle. 1.618... 1.618... anything ◮ Step 2: Perform identifications of the boundary of foliated regions to get a foliation ( F , µ ). 1 2.618... 1.618... 8
Train track to foliation ◮ Step 1: Replace each edge of τ by a foliated Euclidean rectangle. 1.618... 1.618... anything ◮ Step 2: Perform identifications of the boundary of foliated regions to get a foliation ( F , µ ). ◮ We say that τ carries the foliation ( F , µ ). 1 2.618... 1.618... 8
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