Fibrations of 3-manifolds and nowhere continuous functions

Transcription

1 Fibrations of 3-manifolds and nowhere continuous functions Balázs Strenner Georgia Institute of Technology Geometric Topology Seminar Columbia University October 5, 2018

2 Calculus exercise 1

3 Calculus exercise 1 Find a function!: Q R such that

4 Calculus exercise 1 Find a function!: Q R such that 1.! is discontinuous at every x Q.

5 Calculus exercise 1 Find a function!: Q R such that 1.! is discontinuous at every x Q. 2. The limit extension! +, = lim!(+) exists for 1 12 all +, R and! is continuous.

6 Calculus exercise 1 Find a function!: Q R such that 1.! is discontinuous at every x Q. 2. The limit extension! +, = lim!(+) exists for 1 12 all +, R and! is continuous. Solution: For example,! 5 6 = 7 6. Then! = 0.

7 Calculus exercise 2

8 Calculus exercise 2 Let! R $ be a compact, convex polyhedron whose faces are determined by rational equations.!

9 Calculus exercise 2 Let! R $ be a compact, convex polyhedron whose faces are determined by rational equations. Consider a function %: '()(!) Q R.! / Q denotes the rational points of a set / R $.

10 Calculus exercise 2 a Let! R $ be a compact, convex polyhedron whose faces are determined by rational equations.! b Consider a function %: '()(!) Q R. 4 Q denotes the rational points of a set 4 R $. For /, 1 3! Q where the line segment /1 meets '()(!),

11 Calculus exercise 2 Let! R $ be a compact, convex polyhedron whose faces are determined by rational equations. a! t=-1/2 t=0 t=1/2 b Consider a function %: '()(!) Q R.? Q denotes the rational points of a set? R $. For /, 1 3! Q where the line segment /1 meets '()(!), consider 4 5,6 ) = % ) 6;5, defined 9 9 on Q ( 1,1). This is a slice of %.

12 Devil s bowl

13 Devil s bowl

14 A function!: #$%(') Q R is a devil s bowl if '

15 A function!: #$%(') Q R is a devil s bowl if 1.! is discontinuous at every x #$%(') Q. '

16 A function!: #$%(') Q R is a devil s bowl if 1.! is discontinuous at every x #$%(') Q. 2. For every slice 0 1,3, we have ; 0 1,3 (% 4 ) = lim 0 1,3 (%) = 9 9: (<=9 : )(<>9 : ) for some? A > 0. a ' b

17 A function!: #$%(') Q R is a devil s bowl if 1.! is discontinuous at every x #$%(') Q. 2. For every slice 0 1,3, we have ; 0 1,3 (% 4 ) = lim 0 1,3 (%) = 9 9: (<=9 : )(<>9 : ) for some? A > 0. Exercise: Construct a devil s bowl. a ' b

18 A function!: #$%(') Q R is a devil s bowl if 1.! is discontinuous at every x #$%(') Q. 2. For every slice 0 1,3, we have ; 0 1,3 (% 4 ) = lim 0 1,3 (%) = 9 9: (<=9 : )(<>9 : ) for some? A > 0. Exercise: Construct a devil s bowl. Explicit formula? a ' b

19 A function!: #$%(') Q R is a devil s bowl if 1.! is discontinuous at every x #$%(') Q. 2. For every slice 0 1,3, we have ; 0 1,3 (% 4 ) = lim 0 1,3 (%) = 9 9: (<=9 : )(<>9 : ) for some? A > 0. Exercise: Construct a devil s bowl. Solution 1: Enumerate slices and do induction. Explicit formula? a ' b

20 A function!: #$%(') Q R is a devil s bowl if 1.! is discontinuous at every x #$%(') Q. 2. For every slice 0 1,3, we have ; 0 1,3 (% 4 ) = lim 0 1,3 (%) = 9 9: (<=9 : )(<>9 : ) for some? A > 0. Exercise: Construct a devil s bowl. Solution 1: Enumerate slices and do induction. Solution 2: Study fibrations of 3-manifolds. Explicit formula? a ' b

21 3-manifolds

22 3-manifold fibering over the circle. # "! "

23 3-manifold fibering over the circle. # "! " # $ M can fiber in multiple ways.

24 3-manifold fibering over the circle. # "! " # $ M can fiber in multiple ways. How can we distinguish them?

25 3-manifold fibering over the circle. # "! " % = ' ( ) ( Z # $ M can fiber in multiple ways. How can we distinguish them?

26 * & (#) % & / 3-manifold fibering over the circle. 0 "! " # = % & ' & Z 0 1 M can fiber in multiple ways. How can we distinguish them? * 2 (#) % & /

27 primitive (If fibers are connected) * & (#) % & / 3-manifold fibering over the circle. 0 "! " # = % & ' & Z 0 1 M can fiber in multiple ways. How can we distinguish them? primitive (If fibers are connected) * 2 (#) % & /

28 Which elements of! " # correspond to fibrations? Thurston norm $ % on! " #, R

29 Which elements of! " # correspond to fibrations? Thurston norm % & on! " #, R Unit ball: polyhedron $

30 Which elements of! " # correspond to fibrations? Thurston norm $ % on! " #, R Unit ball: polyhedron fibered face: every integral class in this cone encodes a fibration

31 Which elements of! " # correspond to fibrations? Thurston norm $ % on! " #, R Unit ball: polyhedron fibered face: every integral class in this cone encodes a fibration not fibered face: nothing in this cone encodes a fibration

32 Thurston: M is hyperbolic monodromy f is a pseudo-anosov mapping class.

33 Thurston: M is hyperbolic monodromy f is a pseudo-anosov mapping class. pseudo-anosov mapping classes " Q How does the monodromy change as we vary the fibration?

34 Pseudo-Anosov mapping classes

35 Mapping class group: Mod(S)=Homeo + (S)/isotopy

36 Mapping class group: Mod(S)=Homeo + (S)/isotopy Nielsen-Thurston Classification Finite order

37 Mapping class group: Mod(S)=Homeo + (S)/isotopy Nielsen-Thurston Classification U f(u) Finite order Pseudo-Anosov

38 Mapping class group: Mod(S)=Homeo + (S)/isotopy Nielsen-Thurston Classification U f(u) Finite order Pseudo-Anosov Reducible

39 Mapping class group: Mod(S)=Homeo + (S)/isotopy Nielsen-Thurston Classification Stretch factor:! U f(u) Finite order Pseudo-Anosov Reducible

40 " How does! change?

41 Fried ( 82): Let M be a hyperbolic 3-manifold with! " # 2 and fibered face F. ' How does & change?

42 Fried ( 82): Let M be a hyperbolic 3-manifold with! " # 2 and fibered face F. (dim ) =! " (#) 1) ) How does / change?

43 Fried ( 82): Let M be a hyperbolic 3-manifold with! " # 2 and fibered face F. (dim ) =! " (#) 1) The normalized stretch factor function /: ) Q R 4 ) How does 5 change?

44 Fried ( 82): Let M be a hyperbolic 3-manifold with! " # 2 and fibered face F. (dim ) =! " (#) 1) The normalized stretch factor function /: ) Q R 4 / 5 = log ;(5 6 ) ) 5 5 How does ; change?

45 Fried ( 82): Let M be a hyperbolic 3-manifold with! " # 2 and fibered face F. (dim ) =! " (#) 1) The normalized stretch factor function /: ) Q R 4 / 5 = log ;(5 6 ) extends to a convex, continuous function int()) R 4 that goes to infinity at >). ) 5 5 How does ; change?

46 Fried ( 82): Let M be a hyperbolic 3-manifold with! " # 2 and fibered face F. (dim ) =! " (#) 1) The normalized stretch factor function /: ) Q R 4 / 5 = log ;(5 6 ) extends to a convex, continuous function int()) R 4 that goes to infinity at >). 5 Application C DEF,B I B ) 5

47 Arc graph!(#)

48 Arc graph!(#) % & % ( % '

49 Arc graph!(#) % & % &!(#) % ' % ' % ( % (

50 Arc graph!(#) % & % &!(#) % ' % ' % ( % (! # is hyperbolic (Masur-Schleimer)

51 Arc graph!(#) % & % &!(#) % ' % ' % ( % (! # is hyperbolic (Masur-Schleimer) Mod(S) acts by isometries

52 Arc graph!(#) % & % &!(#) % ' % ' % ( % (! # is hyperbolic (Masur-Schleimer) Mod(S) acts by isometries pseudo-anosovs translate along an axis

53 Asymptotic Translation Length ATL $ = lim ) +, - (/, $ ) / ) 2

54 Asymptotic Translation Length ATL $ = lim ) +, - (/, $ ) / ) 2 Futer-Schleimer ( 14): ATL $ is proportional to the height and area of the boundary of the maximal cusp in 3.

55 Theorem (S.): Let M be a complete finite-volume hyperbolic 3-manifold with! " # 2 ' & '

56 Theorem (S.): Let M be a complete finite-volume hyperbolic 3-manifold with! " # 2 and fullypunctured fibered face F. ' & '

57 Theorem (S.): Let M be a complete finite-volume hyperbolic 3-manifold with! " # 2 and fullypunctured fibered face F. The normalized asymptotic translation length in the arc complex function &: ( Q R, & - = - / (- / ) - ( -

58 Theorem (S.): Let M be a complete finite-volume hyperbolic 3-manifold with! " # 2 and fullypunctured fibered face F. The normalized asymptotic translation length in the arc complex function &: ( Q R, & - = - / (- / ) is a devil s bowl. - ( -

59 Theorem (S.): Let M be a complete finite-volume hyperbolic 3-manifold with! " # 2 and fullypunctured fibered face F. The normalized asymptotic translation length in the arc complex function &: ( Q R, & - = - / (- / ) is a devil s bowl. Kin-Shin ( 17): For the curve graph, & - is locally bounded on slices.

60 Theorem (S.): Let M be a complete finite-volume hyperbolic 3-manifold with! " # 2 and fullypunctured fibered face F. The normalized asymptotic translation length in the arc complex function &: ( Q R, & - = - / (- / ) is a devil s bowl. Kin-Shin ( 17): For the curve graph, & - is locally bounded on slices. Application: :,< > <? (Gadre-Tsai 14)

61 Theorem (S.): Let M be a complete finite-volume hyperbolic 3-manifold with! " # 2 and fullypunctured fibered face F. The normalized asymptotic translation length in the arc complex function &: ( Q R, & - = - / (- / ) is a devil s bowl. Kin-Shin ( 17): For the curve graph, & - is locally bounded on slices. Application: :,< > <? (Gadre-Tsai 14) Baik-Shin-Wu ( 18): - / 0 A / is locally bounded on d- dimensional slices.

62 Kin-Shin ( 17): For the curve graph,! " is locally bounded on slices. Application: #$% &'(,*, * - (Gadre-Tsai 14) Baik-Shin-Wu ( 18): ". / #$% ". is locally bounded on d- dimensional slices.

63 Kin-Shin ( 17): For the curve graph,! " is locally bounded on slices. Application: #$% &'(,*, * - (Gadre-Tsai 14) Baik-Shin-Wu ( 18): " / #$% ". is locally bounded on d- dimensional slices. Conjecture (Baik-Shin-Wu): ". / #$% ". is bounded away from 0 on an infinite set.

64 The general theorem

65 Theorem (S.): Let M be a complete finitevolume hyperbolic 3-manifold with! " # 2 and fully-punctured fibered face F.

66 Theorem (S.): Let M be a complete finitevolume hyperbolic 3-manifold with! " # 2 and fully-punctured fibered face F. Let & ' : ) Q R - & '. =. 0 1 "- "' 234(. 0 ) be the 7-adic normalized asymptotic translation length in the arc complex function.

67 Theorem (S.): Let M be a complete finitevolume hyperbolic 3-manifold with! " # 2 and fully-punctured fibered face F. Let & ' : ) Q R - & '. =. 0 1 "- "' 234(. 0 ) be the 7-adic normalized asymptotic translation length in the arc complex function. Let Ω be a 7-dimensional slice of F.

68 Theorem (S.): Let M be a complete finitevolume hyperbolic 3-manifold with! " # 2 and fully-punctured fibered face F. Let & ' : ) Q R - & '. =. 0 "- "' 1 234(. 0 ) be the 7-adic normalized asymptotic translation length in the arc complex function. Let Ω be a 7-dimensional slice of F. Then there is a continuous function 9: :;< Ω R - such that 9 = as =?Ω

69 Theorem (S.): Let M be a complete finitevolume hyperbolic 3-manifold with! " # 2 and fully-punctured fibered face F. Let & ' : ) Q R - & '. =. 0 "- "' 1 234(. 0 ) be the 7-adic normalized asymptotic translation length in the arc complex function. Let Ω be a 7-dimensional slice of F. Then there is a continuous function 9: :;< Ω R - such that 9 = as =?Ω and the set of accumulation points ' F ) in :;< Ω R - is

70 Theorem (S.): Let M be a complete finitevolume hyperbolic 3-manifold with! " # 2 and fully-punctured fibered face F. Let & ' : ) Q R - & '. =. 0 "- "' 1 234(. 0 ) be the 7-adic normalized asymptotic translation length in the arc complex function. Let Ω be a 7-dimensional slice of F. Then there is a continuous function 9: :;< Ω R - such that 9 = as =?Ω and the set of accumulation points ' F ) in :;< Ω R - if 7 = 1

71 Theorem (S.): Let M be a complete finitevolume hyperbolic 3-manifold with! " # 2 and fully-punctured fibered face F. Let & ' : ) Q R - & '. =. 0 "- "' 1 234(. 0 ) be the 7-adic normalized asymptotic translation length in the arc complex function. Let Ω be a 7-dimensional slice of F. Then there is a continuous function 9: :;< Ω R - such that 9 = as =?Ω and the set of accumulation points ' F ) in :;< Ω R - if 7 = 1 The region if 7 2.

72 Corollary: If! 2, then $ % ' is a nowhere continuous function.

73 Corollary: If! 2, then $ % ' is a nowhere continuous function. Corollary (conjecture of Baik-Shin-Wu): $ % ' is bounded away from 0 on infinite sequences.

74 Corollary: If! 2, then $ % ' is a nowhere continuous function. Corollary (conjecture of Baik-Shin-Wu): $ % ' is bounded away from 0 on infinite sequences. There is a formula for (, but it is complicated.

75 Corollary: If! 2, then $ % ' is a nowhere continuous function. Corollary (conjecture of Baik-Shin-Wu): $ % ' is bounded away from 0 on infinite sequences. There is a formula for (, but it is complicated. Question: Is the bounding function ( always convex?

76 Theorem 3 (S.): Let!, ", #, Ω and % as in the previous theorem.

77 Theorem 3 (S.): Let!, ", #, Ω and % as in the previous theorem. Suppose Ω is a simplex with vertices & ',, & *+' and define the parametrization *+' % - ',, - *+' = % / 01' - 0 & 0 by { - ',, - *+' : 0 < - 0, *+' 01' - 0 = 1}, the interior of the standard simplex.

78 Theorem 3 (S.): Let!, ", #, Ω and % as in the previous theorem. Suppose Ω is a simplex with vertices & ',, & *+' and define the parametrization *+' % - ',, - *+' = % / 01' - 0 & 0 by { - ',, - *+' : 0 < - 0, *+' 01' - 0 = 1}, the interior of the standard simplex. Let Σ be the subspace spanned by Ω and let Γ = Σ < ' (!; Z) be the integral lattice in Σ.

79 Theorem 3 (S.): Let!, ", #, Ω and % as in the previous theorem. Suppose Ω is a simplex with vertices & ',, & *+' and define the parametrization *+' % - ',, - *+' = % / 01' - 0 & 0 by { - ',, - *+' : 0 < - 0, *+' 01' - 0 = 1}, the interior of the standard simplex. Let Σ be the subspace spanned by Ω and let Γ = Σ < ' (!; Z) be the integral lattice in Σ. Then % - ',, - *+' = A BCD E/G H A *! BCD E/ K L,,K AML Z OPL where R * is a constant depending only on #. AML QO

80 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior.

81 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior. Example (d=2):

82 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior. Example (d=2):

83 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior. Example (d=2):

84 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior. Example (d=2):

85 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior. Example (d=2):?

86 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior. Example (d=2):? Fact 1:! " 1 for all,.

87 The constant! "! " = the smallest possible volume of a simplex # in R " with the property that each of its affine copy %# + ' (% > 1) contains a point of Z " in its interior. Example (d=2):? Fact 1:! " 1 for all,. Fact 2:!. = 1.

88 Proof of the the main theorem

89 Frobenius coin problem Let A = {$ %,, $ ( } be a set of relatively prime positive integers.

90 Frobenius coin problem Let A = {$ %,, $ ( } be a set of relatively prime positive integers. The Frobenius number of the set A is the largest integer that does not arise as < % $ % + + < ( $ ( for nonnegative integers <?.

91 Frobenius coin problem Let A = {$ %,, $ ( } be a set of relatively prime positive integers. The Frobenius number of the set A is the largest integer that does not arise as < % $ % + + < ( $ ( for nonnegative integers <?. Examples: 1. ABCD 3,5 = 7

92 Frobenius coin problem Let A = {$ %,, $ ( } be a set of relatively prime positive integers. The Frobenius number of the set A is the largest integer that does not arise as < % $ % + + < ( $ ( for nonnegative integers <?. Examples: 1. ABCD 3,5 = 7 2. ABCD $, D = $D $ D

93 Frobenius numbers of cones Let C = # $,, # ' R) R + be a cone with nonempty interior. Let,: Z + Z be a surjective homomorphism.

94 Frobenius numbers of cones Let C = # $,, # ' R) R + be a cone with nonempty interior. Let,: Z + Z be a surjective homomorphism. The Frobenius number of C with respect to, is the largest integer that is not contained in,(1 Z + ).

95 Frobenius numbers of cones Let C = # $,, # ' R) R + be a cone with nonempty interior. Let,: Z + Z be a surjective homomorphism. The Frobenius number of C with respect to, is the largest integer that is not contained in,(1 Z + ) Example:

96 Frobenius numbers of cones Let C = # $,, # ' R) R + be a cone with nonempty interior. Let,: Z + Z be a surjective homomorphism. The Frobenius number of C with respect to, is the largest integer that is not contained in,(1 Z + ) Example: Frob =

97 Main steps of the proof 1. ATL(x ) = average weight of cycles in a finite weighted graph W(x )

98 Main steps of the proof 1. ATL(x ) = average weight of cycles in a finite weighted graph W(x ) 2. Weights Frobenius numbers of a cone C in H # $; Z with respect to x ( * # $; Z.

99 Main steps of the proof 1. ATL(x ) = average weight of cycles in a finite weighted graph W(x ) 2. Weights Frobenius numbers of a cone C in H # $; Z with respect to x ( * # $; Z. 3. Understand the way these Frobenius numbers change when x ( * # $; Z is varied.

100 Main technical tool veering triangulations

101 Main technical tool veering triangulations Introduced by Agol (2011)

102 Main technical tool veering triangulations Introduced by Agol (2011) Alternative viewpoint by Guéritaud (2016)

103 Main technical tool veering triangulations Introduced by Agol (2011) Alternative viewpoint by Guéritaud (2016) Further studied by Minsky-Taylor (2017)

104 Questions 1. Is the bounding function! convex? 2. What is the value of " # when $ 2? 3. Do our results generalize to the curve complex? 4. Is the fully-punctured hypothesis necessary?