(Ortho)spectra and identities TIT 2012

Transcription

1 (Ortho)spectra and identities TIT 2012 Greg McShane September 13, 2012

2 Nightmare Bad dreams...

3 Nightmare: forgot the plane

4 Nightmare: forgot the train

5 Nightmare: forgot the exam

6 Nightmare Forgotten to thank you for...

7 Part I Introduction

8 Surfaces Σ is a surface of finite type totally geodesic boundary finite volume hyperbolic structure

9 Surfaces Σ is a surface of finite type totally geodesic boundary finite volume hyperbolic structure Figure: My favorite surface

10 Surfaces Σ is a surface of finite type totally geodesic boundary finite volume hyperbolic structure Figure: My favorite surface π 1 (Σ) Γ < hyperbolic isometries PSL(2, R).

11 Surfaces Σ is a surface of finite type totally geodesic boundary finite volume hyperbolic structure Figure: My favorite surface π 1 (Σ) Γ < hyperbolic isometries PSL(2, R). Λ = limit set of Γ, H

12 Surfaces Σ is a surface of finite type totally geodesic boundary finite volume hyperbolic structure Figure: My favorite surface π 1 (Σ) Γ < hyperbolic isometries PSL(2, R). Λ = limit set of Γ, H Σ = Convex hull(λ)/γ

13 Convex hull Figure: Convex hullof Λ

14 Spectra Closed geodesic definition shortest path in [γ] γ 1 Γ length tr γ = 2 cosh( 1 2 l(γ))

15 Spectra Closed geodesic Simple closed geodesic definition length shortest path in [γ] γ 1 Γ same as above tr γ = 2 cosh( 1 2 l(γ)) + no self intersection same as above

16 Spectra Closed geodesic Simple closed geodesic Ortho geodesic definition length shortest path in [γ] γ 1 Γ same as above tr γ = 2 cosh( 1 2 l(γ)) + no self intersection same as above α shortest chord joins 2 geodesic see below boundary components tanh 2 is cross ratio

17 Orthogeodesic Ortho geodesic is the common perpendicular to a pair of geodesics [α], [δ] Σ.

18 Orthogeodesic Ortho geodesic is the common perpendicular to a pair of geodesics [α], [δ] Σ. Figure: Convex hull, lift of ortho geodesic α

19 Orthogeodesic Ortho geodesic is the common perpendicular to a pair of geodesics [α], [δ] Σ. Figure: Convex hull, lift of ortho geodesic α If α ±, δ ± are the fixed points on H then (α δ )(α + δ + ) (α δ + )(α + δ ) = tanh2 ( 1 2 l(α ))

20 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian}

21 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics}

22 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics} Ortho spectrum = {lengths of ortho geodesics}

23 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics} Ortho spectrum = {lengths of ortho geodesics} What geometric quantities are functions of the spectrum? metric on Σ up to isometry.

24 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics} Ortho spectrum = {lengths of ortho geodesics} What geometric quantities are functions of the spectrum? metric on Σ up to isometry. injectivity radius

25 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics} Ortho spectrum = {lengths of ortho geodesics} What geometric quantities are functions of the spectrum? metric on Σ up to isometry. injectivity radius Vol(Σ)

26 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics} Ortho spectrum = {lengths of ortho geodesics} What geometric quantities are functions of the spectrum? metric on Σ up to isometry. injectivity radius Vol(Σ) Vol( Σ)

27 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics} Ortho spectrum = {lengths of ortho geodesics} What geometric quantities are functions of the spectrum? metric on Σ up to isometry. injectivity radius Vol(Σ) Vol( Σ) δ = Hausdorff dimension of the limit set.

28 Spectra What are the spectra of a manifold? Spectrum = {eigenvalues of Lapacian} Length spectrum = {lengths of closed geodesics} Ortho spectrum = {lengths of ortho geodesics} What geometric quantities are functions of the spectrum? metric on Σ up to isometry. injectivity radius Vol(Σ) Vol( Σ) δ = Hausdorff dimension of the limit set.

29 Spectrum and geometry for a compact surface For a compact surface H/Γ: (Weyl) Spectrum of Laplacian determines area N Γ (t) := {eigenvalues of H/Γ < t} Vol(H/Γ) t 4π

30 Spectrum and geometry for a compact surface For a compact surface H/Γ: (Weyl) Spectrum of Laplacian determines area N Γ (t) := {eigenvalues of H/Γ < t} Vol(H/Γ) t 4π (Huber, Selberg) Length spectrum determines spectrum of the Laplacian

31 Spectrum and geometry for a compact surface For a compact surface H/Γ: (Weyl) Spectrum of Laplacian determines area N Γ (t) := {eigenvalues of H/Γ < t} Vol(H/Γ) t 4π (Huber, Selberg) Length spectrum determines spectrum of the Laplacian and vice versa

32 Spectrum and geometry for a compact surface For a compact surface H/Γ: (Weyl) Spectrum of Laplacian determines area N Γ (t) := {eigenvalues of H/Γ < t} Vol(H/Γ) t 4π (Huber, Selberg) Length spectrum determines spectrum of the Laplacian and vice versa (Margulis/Sullivan) Length spectrum determines Hausdorff dimension N Γ (t) := {primitive geodesics l(α) < t} eδt δt.

33 Spectrum and geometry for a compact surface For a compact surface H/Γ: (Weyl) Spectrum of Laplacian determines area N Γ (t) := {eigenvalues of H/Γ < t} Vol(H/Γ) t 4π (Huber, Selberg) Length spectrum determines spectrum of the Laplacian and vice versa (Margulis/Sullivan) Length spectrum determines Hausdorff dimension N Γ (t) := {primitive geodesics l(α) < t} eδt δt. (Wolpert) Length spectrum determines isometry type of the surface up to finitely many choices.

34 Spectrum and geometry for a compact surface For a compact surface H/Γ: (Weyl) Spectrum of Laplacian determines area N Γ (t) := {eigenvalues of H/Γ < t} Vol(H/Γ) t 4π (Huber, Selberg) Length spectrum determines spectrum of the Laplacian and vice versa (Margulis/Sullivan) Length spectrum determines Hausdorff dimension N Γ (t) := {primitive geodesics l(α) < t} eδt δt. (Wolpert) Length spectrum determines isometry type of the surface up to finitely many choices. Examples of isospectral pairs (Vigneras, Sunada).

35 Trace formula h even function, satisfying a growth condition ĥ Fourier transform

36 Trace formula h even function, satisfying a growth condition ĥ Fourier transform where n h(λ n ) = Vol(H/Γ) 4π + [γ] R rh(r) tanh(πr)dr 2l(γ) sinh( 1 2 l(γ))ĥ(l(γ))

37 Trace formula h even function, satisfying a growth condition ĥ Fourier transform where n h(λ n ) = Vol(H/Γ) 4π + [γ] R rh(r) tanh(πr)dr 2l(γ) sinh( 1 2 l(γ))ĥ(l(γ)) λ n are the eigenvalues of the Laplacian. l(γ) is the length of the geodesic in the homotopy class [γ]

38 Ortho Spectra McShane Parker Redfern, 93 Drawing limit sets using finite state automata.

39 Ortho Spectra McShane Parker Redfern, 93 Drawing limit sets using finite state automata. Definition Circle packing iff complement of the circles has empty interior.

40 Ortho Spectra Orthogeodesic = common perpendicular between circles. Figure: David Wright s circle packing

41 Ortho Spectra Orthogeodesic = common perpendicular between circles. Figure: Curt McMullen s circle packing limit set

42 Ortho Spectra (Basmajian, Parker) Has the same critical exponent δ as the full Poincaré series inf{s, g Γ e sd(x,g(x)) < }

43 Ortho Spectra (Basmajian, Parker) Has the same critical exponent δ as the full Poincaré series inf{s, g Γ e sd(x,g(x)) < } inf{s, { [γ] [Γ] e sl(γ) < }

44 Ortho Spectra (Basmajian, Parker) Has the same critical exponent δ as the full Poincaré series inf{s, g Γ e sd(x,g(x)) < } inf{s, { [γ] [Γ] e sl(γ) < } inf{s, { α e sl(α ) ) < }

45 Ortho Spectra (Basmajian, Parker) Has the same critical exponent δ as the full Poincaré series inf{s, g Γ e sd(x,g(x)) < } inf{s, { [γ] [Γ] e sl(γ) < } inf{s, { α e sl(α ) ) < } By Sullivan δ the Hausdorff dimension of the limit set for geometrically finite groups.

46 Ortho Spectra (Basmajian, Parker) Has the same critical exponent δ as the full Poincaré series inf{s, g Γ e sd(x,g(x)) < } inf{s, { [γ] [Γ] e sl(γ) < } inf{s, { α e sl(α ) ) < } By Sullivan δ the Hausdorff dimension of the limit set for geometrically finite groups. (Roblin, Hee Oh, Kontorovich) Has the same asymptotic as the length spectrum. N(t) := {ortho geodesics l(α ) < t} eδt δt.

47 Part II Identities

48 ...a complicated way to calculate volume.

49 Basmajian Identity Theorem (1992) Σ hyperbolic surface with geodesic boundary a geodesic δ ( ) 2 sinh 1 1 sinh(l(α = l(δ) ) α

50 Basmajian Identity Theorem (1992) Σ hyperbolic surface with geodesic boundary a geodesic δ ( ) 2 sinh 1 1 sinh(l(α = l(δ) ) α M hyperbolic n-manifold with geodesic boundary ( )) Vol n 1 (Ball radius = sinh 1 1 sinh(l(α = Vol n 1 ( M) ) α

51 Bridgeman-Kahn Identity Theorem (2008) Σ hyperbolic surface with geodesic boundary δ 2πVol(Σ) = 8 ( ) 1 L cosh 2 (l(α α )/2)

52 Bridgeman-Kahn Identity Theorem (2008) Σ hyperbolic surface with geodesic boundary δ 2πVol(Σ) = 8 ( ) 1 L cosh 2 (l(α α )/2) Dilogarithm Li 2 (z) = z k k 2 = z 0 log(1 x) dx x

53 Bridgeman-Kahn Identity Theorem (2008) Σ hyperbolic surface with geodesic boundary δ 2πVol(Σ) = 8 ( ) 1 L cosh 2 (l(α α )/2) Dilogarithm Li 2 (z) = z k k 2 = z 0 log(1 x) dx x Roger s dilogarithm L(x) = Li 2 (x) + 1 log x log(1 x), x < 1. 2 L (x) = 1 ( log(1 x) + log(x) ). 2 x 1 x

54 Bridgeman-Kahn Identity in general Theorem Σ hyperbolic surface with geodesic boundary 2πVol(Σ) = 8 ( ) 1 L cosh 2 (l(α α )/2)

55 Bridgeman-Kahn Identity in general Theorem Σ hyperbolic surface with geodesic boundary 2πVol(Σ) = 8 ( ) 1 L cosh 2 (l(α α )/2) Exist F n such that for any hyperbolic n-manifold M with geodesic boundary Vol(M) = α F n (l(α )) the volume of M is equal to the sum of the values of F n on the orthospectrum of M.

56 Bridgeman-Kahn Identity in general Theorem Σ hyperbolic surface with geodesic boundary 2πVol(Σ) = 8 ( ) 1 L cosh 2 (l(α α )/2) Exist F n such that for any hyperbolic n-manifold M with geodesic boundary Vol(M) = α F n (l(α )) the volume of M is equal to the sum of the values of F n on the orthospectrum of M. + integral formula for F n in terms of elementary functions.

57 Identity for closed simple geodesics Σ has a one-holed torus, boundary component of length l(δ) = 0 α simple. α e l(α) = 1

58 Identity for closed simple geodesics Σ has a one-holed torus, boundary component of length l(δ) = 0 α simple. α e l(α) = 1 Basmajian in dimension 3 (Calegari) 4 e (l(α ) 1 = χ( M) α

59 Identity for closed simple geodesics Σ has a one-holed torus, boundary component of length l(δ) = 0 α simple. α e l(α) = 1 Basmajian in dimension 3 (Calegari) 4 e (l(α ) 1 = χ( M) α Bridgeman-Kahn in dimension 3 (Masai) l(α ) + 1 e (l(α ) 1 = 2πVol(M) α

60 Resume: Ortho spectrum... Determines volume of M and volume of M

61 Resume: Ortho spectrum... Determines volume of M and volume of M Basmajian in dimension 3 (Calegari) 4 e (l(α ) 1 = χ( M) α

62 Resume: Ortho spectrum... Determines volume of M and volume of M Basmajian in dimension 3 (Calegari) 4 e (l(α ) 1 = χ( M) α Bridgeman-Kahn in dimension 3 (Masai) l(α ) + 1 e (l(α ) 1 = 2πVol(M) α

63 Forgotten identities... Bowditch

64 Forgotten identities... Bowditch Sakuma

65 Forgotten identities... Bowditch Sakuma Mirzakhani

66 Forgotten identities... Bowditch Sakuma Mirzakhani Labourie-McShane

67 Forgotten identities... Bowditch Sakuma Mirzakhani Labourie-McShane Luo-Tan

68 Forgotten identities... Bowditch Sakuma Mirzakhani Labourie-McShane Luo-Tan Kim-Tan

69 Forgotten identities... Bowditch Sakuma Mirzakhani Labourie-McShane Luo-Tan Kim-Tan Huay

70 Forgotten identities... Bowditch Sakuma Mirzakhani Labourie-McShane Luo-Tan Kim-Tan Huay and anyone else...

71 Part III Applications

72 Dilogarithm identities For Σ H an ideal n-gon:

73 Dilogarithm identities For Σ H an ideal n-gon: hyperbolic area (n 2)π

74 Dilogarithm identities For Σ H an ideal n-gon: hyperbolic area (n 2)π vertices x i, i = 1,..., n

75 Dilogarithm identities For Σ H an ideal n-gon: hyperbolic area (n 2)π vertices x i, i = 1,..., n l ij = length of the orthogeodesic x i x i+1 x j x j+1 cosh 2 ( 1 2 l ij) = [x i, x i+1, x j, x j+1 ]

76 Dilogarithm identities For Σ H an ideal n-gon: hyperbolic area (n 2)π vertices x i, i = 1,..., n l ij = length of the orthogeodesic x i x i+1 x j x j+1 cosh 2 ( 1 2 l ij) = [x i, x i+1, x j, x j+1 ]

77 Dilogarithm identities For Σ H an ideal n-gon: hyperbolic area (n 2)π vertices x i, i = 1,..., n l ij = length of the orthogeodesic x i x i+1 x j x j+1 cosh 2 ( 1 2 l ij) = [x i, x i+1, x j, x j+1 ] Theorem (Bridgeman s Length Spectrum Identity) L([x i, x i+1, x j, x j+1 ]) = α i,j ( ) 1 L cosh 2 = (l α /2) (n 3)π2 6

78 Dilogarithm identities Theorem (Bridgeman s Length Spectrum Identity) L([x i, x i+1, x j, x j+1 ]) = α i,j ( ) 1 L cosh 2 = (l α /2) (n 3)π2 6

79 Dilogarithm identities Theorem (Bridgeman s Length Spectrum Identity) L([x i, x i+1, x j, x j+1 ]) = α i,j ( ) 1 L cosh 2 = (l α /2) (n 3)π2 6 This is a finite summation relation associated relations give identities for dilogarithms including the classical identities:

80 Dilogarithm identities Theorem (Bridgeman s Length Spectrum Identity) L([x i, x i+1, x j, x j+1 ]) = α i,j ( ) 1 L cosh 2 = (l α /2) (n 3)π2 6 This is a finite summation relation associated relations give identities for dilogarithms including the classical identities: EULER L(x) + L(1 x) = L(1) = π2 6

81 Dilogarithm identities Theorem (Bridgeman s Length Spectrum Identity) L([x i, x i+1, x j, x j+1 ]) = α i,j ( ) 1 L cosh 2 = (l α /2) (n 3)π2 6 This is a finite summation relation associated relations give identities for dilogarithms including the classical identities: EULER L(x) + L(1 x) = L(1) = π2 6 ABEL L(x) + L(y) = L(xy) + L( x(1 y) y(1 x) 1 xy ) + L( 1 xy )

82 Dilogarithm identities Theorem (Bridgeman s Length Spectrum Identity) L([x i, x i+1, x j, x j+1 ]) = α i,j ( ) 1 L cosh 2 = (l α /2) (n 3)π2 6 This is a finite summation relation associated relations give identities for dilogarithms including the classical identities: EULER L(x) + L(1 x) = L(1) = π2 6 ABEL L(x) + L(y) = L(xy) + L( x(1 y) y(1 x) 1 xy ) + L( 1 xy )

83 Euler reflection L(x) + L(1 x) = L(1) = π2 6

84 Euler reflection L(x) + L(1 x) = L(1) = π2 6 Proof: The ideal quadrilateral has two ortho geodesics

85 Euler reflection L(x) + L(1 x) = L(1) = π2 6 Proof: The ideal quadrilateral has two ortho geodesics L(cross ratio 1) + L(cross ratio 2) = (4 3)π2. 6

86 Euler reflection L(x) + L(1 x) = L(1) = π2 6 Proof: The ideal quadrilateral has two ortho geodesics L(cross ratio 1) + L(cross ratio 2) = (4 3)π2. 6 vertices 0 < x < 1 <

87 Euler reflection L(x) + L(1 x) = L(1) = π2 6 Proof: The ideal quadrilateral has two ortho geodesics L(cross ratio 1) + L(cross ratio 2) = (4 3)π2. 6 vertices 0 < x < 1 < two ortho geodesics joining the sides: x, 1 to, 0 0, x to 1,

88 Euler reflection L(x) + L(1 x) = L(1) = π2 6 Proof: The ideal quadrilateral has two ortho geodesics L(cross ratio 1) + L(cross ratio 2) = (4 3)π2. 6 vertices 0 < x < 1 < two ortho geodesics joining the sides: x, 1 to, 0 0, x to 1, So cross ratio 1 = x 0 x = x cross ratio 2 = x 1 x = 1 x

89 Part IV Proofs

90 Decompositions Given an identity : what is the associated geometric decomposition?

91 Decompositions Given an identity : what is the associated geometric decomposition? Decomposition: some space X = ( {geometric pieces}) {negligible}

92 Decompositions Given an identity : what is the associated geometric decomposition? Decomposition: some space X = ( {geometric pieces}) {negligible} X = H, negligible = Λ

93 Decompositions Given an identity : what is the associated geometric decomposition? Decomposition: some space X = ( {geometric pieces}) {negligible} X = H, negligible = Λ X = unit tangent bundle Σ, negligible = geodesics that stay in convex core.

94 Decompositions Given an identity : what is the associated geometric decomposition? Decomposition: some space X = ( {geometric pieces}) {negligible} X = H, negligible = Λ X = unit tangent bundle Σ, negligible = geodesics that stay in convex core. X = unit tangent bundle convex hull of Λ, negligible = geodesics that stay in convex hull.

95 Negligible sets Theorem Γ is geometrically finite, and Λ c then the limit set Λ has measure zero.

96 Negligible sets Theorem Γ is geometrically finite, and Λ c then the limit set Λ has measure zero. Definition γ v := geodesic parametrized by arclength, γ v (0) = v

97 Negligible sets Theorem Γ is geometrically finite, and Λ c then the limit set Λ has measure zero. Definition γ v := geodesic parametrized by arclength, γ v (0) = v Proposition (almost all geodesics escape) If Λ c then the set ω + (Λ) := { unit vectors v UT H, γ v ( ) Λ} is measure zero.

98 Negligible sets Theorem Γ is geometrically finite, and Λ c then the limit set Λ has measure zero. Definition γ v := geodesic parametrized by arclength, γ v (0) = v Proposition (almost all geodesics escape) If Λ c then the set ω + (Λ) := { unit vectors v UT H, γ v ( ) Λ} is measure zero. Proof:

99 Negligible sets Theorem Γ is geometrically finite, and Λ c then the limit set Λ has measure zero. Definition γ v := geodesic parametrized by arclength, γ v (0) = v Proposition (almost all geodesics escape) If Λ c then the set ω + (Λ) := { unit vectors v UT H, γ v ( ) Λ} is measure zero. Proof: The unit tangent bundle H S 1 H H = {(x, θ)} Apply Fubini s Theorem to characteristic function of ω + (Λ).

100 Negligible sets Theorem Γ is geometrically finite, and Λ c then the limit set Λ has measure zero. Definition γ v := geodesic parametrized by arclength, γ v (0) = v Proposition (almost all geodesics escape) If Λ c then the set ω + (Λ) := { unit vectors v UT H, γ v ( ) Λ} is measure zero. Proof: The unit tangent bundle H S 1 H H = {(x, θ)} Apply Fubini s Theorem to characteristic function of ω + (Λ).

101 Decomposition from convex hull boundary Proposition (all geodesics escape) If Λ c then the set ω + (Λ) of unit vectors v such that γ v stays in the convex hull is measure 0.

102 Decomposition from convex hull boundary Proposition (all geodesics escape) If Λ c then the set ω + (Λ) of unit vectors v such that γ v stays in the convex hull is measure 0. The generic geodesic escapes the convex hull by crossing some pair of planes bounding disjoint round discs in Ω.

103 Decomposition from convex hull boundary Proposition (all geodesics escape) If Λ c then the set ω + (Λ) of unit vectors v such that γ v stays in the convex hull is measure 0. The generic geodesic escapes the convex hull by crossing some pair of planes bounding disjoint round discs in Ω. Decomposition of (a subset of full measure of) unit tangent bundle labelled by pairs of planes.

104 Decomposition from convex hull boundary Proposition (all geodesics escape) If Λ c then the set ω + (Λ) of unit vectors v such that γ v stays in the convex hull is measure 0. The generic geodesic escapes the convex hull by crossing some pair of planes bounding disjoint round discs in Ω. Decomposition of (a subset of full measure of) unit tangent bundle labelled by pairs of planes. Decomposition of (a subset of full measure of) unit tangent bundle labelled by ortho geodesics.

105 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic.

106 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic. S M H n is a copy of H n 1.

107 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic. S M H n is a copy of H n 1. S Λ H n is a round sphere.

108 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic. S M H n is a copy of H n 1. S Λ H n is a round sphere. complement of S =pair of hemispheres in H n

109 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic. S M H n is a copy of H n 1. S Λ H n is a round sphere. complement of S =pair of hemispheres in H n one of the hemispheres is entirely in the regular set Ω. remaining components of Ω yields a sphere packing of the other hemispheres.

110 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic. S M H n is a copy of H n 1. S Λ H n is a round sphere. complement of S =pair of hemispheres in H n one of the hemispheres is entirely in the regular set Ω. remaining components of Ω yields a sphere packing of the other hemispheres.

111 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic.

112 Caligari s Chimneys M = compact hyperbolic n-manifold, M = S totally geodesic.

113 Caligari s Chimneys Definition A chimney is isometric to convex hull of

114 Caligari s Chimneys Definition A chimney is isometric to convex hull of a plane bounding a component of Ω + the image of the plane under retraction to S.

115 Caligari s Chimneys Definition A chimney is isometric to convex hull of a plane bounding a component of Ω + the image of the plane under retraction to S.

116 Caligari s Chimneys Definition A chimney is isometric to convex hull of a plane bounding a component of Ω + the image of the plane under retraction to S. Each ortho geodesic in α M lifts to a geodesic arc with initial point in S and endpoint in some plane.

117 Caligari s Chimneys Definition A chimney is isometric to convex hull of a plane bounding a component of Ω + the image of the plane under retraction to S. Each ortho geodesic in α M lifts to a geodesic arc with initial point in S and endpoint in some plane. Core of a chimney = α Top of a chimney is the plane. Base a round disk in the surface S.

118 Basmajian identity Collection of bases of the chimneys forms circle packing of S. M is geometrically finite so complement = set of measure zero.

119 Basmajian identity Collection of bases of the chimneys forms circle packing of S. M is geometrically finite so complement = set of measure zero. M = ( α base of chimney) projection of Λ

120 Basmajian identity Collection of bases of the chimneys forms circle packing of S. M is geometrically finite so complement = set of measure zero. M = ( α base of chimney) projection of Λ Vol( M) = α Vol(base of chimney)

121 Basmajian identity Collection of bases of the chimneys forms circle packing of S. M is geometrically finite so complement = set of measure zero. M = ( α base of chimney) projection of Λ Vol( M) = α Vol(base of chimney) Bridgeman-Kahn will discuss in detail later, but roughly:

122 Basmajian identity Collection of bases of the chimneys forms circle packing of S. M is geometrically finite so complement = set of measure zero. M = ( α base of chimney) projection of Λ Vol( M) = α Vol(base of chimney) Bridgeman-Kahn will discuss in detail later, but roughly: TM = ( α B(α )) tangents to geodesics with endpts Λ

123 Basmajian identity Collection of bases of the chimneys forms circle packing of S. M is geometrically finite so complement = set of measure zero. M = ( α base of chimney) projection of Λ Vol( M) = α Vol(base of chimney) Bridgeman-Kahn will discuss in detail later, but roughly: TM = ( α B(α )) tangents to geodesics with endpts Λ Vol(UTM) = α Vol(UB(α ))

124 Part V Equidecomposability

125 Bridgeman and Calegari decompositions ˆδ ˆα Figure: geodesics escaping p(v) is a big dot.

126 Bridgeman and Calegari decompositions ˆδ ˆα Figure: geodesics escaping p(v) is a big dot. Subsets of the unit tangent bundle of the ideal quadrilateral Q(α ). v tangent bundle p(v) H is the base of the vector.

127 Bridgeman and Calegari decompositions ˆδ ˆα Figure: geodesics escaping p(v) is a big dot. Subsets of the unit tangent bundle of the ideal quadrilateral Q(α ). v tangent bundle p(v) H is the base of the vector. Bridgeman s set B(α ) is the set of vectors v tangent to geodesic segments joining ˆα to ˆδ 1. the ray γ v (R + ) meets ˆδ, 2. the ray γ v (R ) meets ˆα.

128 Bridgeman and Calegari decompositions ˆδ ˆα Figure: geodesics escaping p(v) is a big dot. Subsets of the unit tangent bundle of the ideal quadrilateral Q(α ). v tangent bundle p(v) H is the base of the vector. Bridgeman s set B(α ) is the set of vectors v tangent to geodesic segments joining ˆα to ˆδ 1. the ray γ v (R + ) meets ˆδ, 2. the ray γ v (R ) meets ˆα. Calegari s set C(α ) is the set of vectors v such that 1. the ray γ v (R + ) meets ˆδ, 2. the base p(v) is in the chimney of the quadrilateral.

129 Identities Theorem (Bridgeman-Kahn, Calegari) Vol(S n 1 )Vol(M) = Vol(UTM)

130 Identities Theorem (Bridgeman-Kahn, Calegari) Vol(S n 1 )Vol(M) = Vol(UTM) = α Vol(UB(α )) = α Vol(UC(α ))

131 Identities Theorem (Bridgeman-Kahn, Calegari) Vol(S n 1 )Vol(M) = Vol(UTM) = α Vol(UB(α )) = α Vol(UC(α )) Proof: TM has a decomposition into pieces (either B(α )s or C(α )s) one for each orthogeodesic α the complement consists of v, γ v ( ) Λ so is measure 0.

132 Identities Theorem (Bridgeman-Kahn, Calegari) Vol(S n 1 )Vol(M) = Vol(UTM) = α Vol(UB(α )) = α Vol(UC(α )) Proof: TM has a decomposition into pieces (either B(α )s or C(α )s) one for each orthogeodesic α the complement consists of v, γ v ( ) Λ so is measure 0. Theorem (M.) For all n 2 Vol(UB(α )) = Vol(UC(α )) Case n = 2, Calegari by direct computation.

133 Equidecomposability Let X, Y metric space are equidecomposable iff there exists

134 Equidecomposability Let X, Y metric space are equidecomposable iff there exists decompositions X = N k=0 X k, Y = N k=0 Y k

135 Equidecomposability Let X, Y metric space are equidecomposable iff there exists decompositions X = N k=0 X k, Y = N k=0 Y k isometries φ k such that Y k = φ k (X k ).

136 Equidecomposability Let X, Y metric space are equidecomposable iff there exists decompositions X = N k=0 X k, Y = N k=0 Y k isometries φ k such that Y k = φ k (X k ). Let X, Y measure space are countable equidecomposable iff there exists

137 Equidecomposability Let X, Y metric space are equidecomposable iff there exists decompositions X = N k=0 X k, Y = N k=0 Y k isometries φ k such that Y k = φ k (X k ). Let X, Y measure space are countable equidecomposable iff there exists measurable decompositions

138 Equidecomposability Let X, Y metric space are equidecomposable iff there exists decompositions X = N k=0 X k, Y = N k=0 Y k isometries φ k such that Y k = φ k (X k ). Let X, Y measure space are countable equidecomposable iff there exists measurable decompositions X = k=0 X k, Y = k=0 Y k isometries φ k such that Y k = φ k (X k ).

139 Equidecomposability Let X, Y metric space are equidecomposable iff there exists decompositions X = N k=0 X k, Y = N k=0 Y k isometries φ k such that Y k = φ k (X k ). Let X, Y measure space are countable equidecomposable iff there exists measurable decompositions X = k=0 X k, Y = k=0 Y k isometries φ k such that Y k = φ k (X k ). The Haar measure is countably additive and so we can conclude that X and Y have the same volume provided the pieces X k, Y k are measurable,

140 Equidecomposability, dimension 2 A := hyperbolic isometry pairing the sides of the chimney

141 Equidecomposability, dimension 2 A := hyperbolic isometry pairing the sides of the chimney ˆδ A(γ) γ A 2 (γ) ˆα A(ˆα) A 2 (ˆα) Figure: Covering of quadrilateral by A k translates of chimney

142 Equidecomposability, dimension 2 A := hyperbolic isometry pairing the sides of the chimney ˆδ A(γ) γ A 2 (γ) ˆα A(ˆα) A 2 (ˆα) Figure: Covering of quadrilateral by A k translates of chimney X k := {v B(α ), A k (p(v)) chimney} Y k := {v C(α ), γ v (R ) A k (ˆα) }.

143 Equidecomposability, dimension 2 A := hyperbolic isometry pairing the sides of the chimney ˆδ A(γ) γ A 2 (γ) ˆα A(ˆα) A 2 (ˆα) Figure: Covering of quadrilateral by A k translates of chimney X k := {v B(α ), A k (p(v)) chimney} Y k := {v C(α ), γ v (R ) A k (ˆα) }. Easy to see Y k = A k (X k ).

144 Visualising this Surface obtained from a chimney by identifying sides. Figure: geodesic on a crown of area π

145 Visualising this Surface obtained from a chimney by identifying sides. Figure: geodesic on a crown of area π Vol(UB(α )) = π Probability a geodesic hits top and bottom

146 Visualising this Surface obtained from a chimney by identifying sides. Figure: geodesic on a crown of area π Vol(UB(α )) = π Probability a geodesic hits top and bottom Vol(UC(α )) = π Probability a geodesic hits top

147 Visualising this Surface obtained from a chimney by identifying sides. Figure: geodesic on a crown of area π Vol(UB(α )) = π Probability a geodesic hits top and bottom Vol(UC(α )) = π Probability a geodesic hits top No geodesic bigons these are the same

148 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney.

149 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing.

150 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing. Instead we choose A k, k = 1, 2,... isometries such that:

151 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing. Instead we choose A k, k = 1, 2,... isometries such that: A k ( S) = S

152 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing. Instead we choose A k, k = 1, 2,... isometries such that: A k ( S) = S {A k (chimney base)} k is a (locally finite) cover of the plane S

153 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing. Instead we choose A k, k = 1, 2,... isometries such that: A k ( S) = S {A k (chimney base)} k is a (locally finite) cover of the plane S

154 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing. Instead we choose A k, k = 1, 2,... isometries such that: A k ( S) = S {A k (chimney base)} k is a (locally finite) cover of the plane S X k := {v B(α ), A k (p(v)) chimney}

155 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing. Instead we choose A k, k = 1, 2,... isometries such that: A k ( S) = S {A k (chimney base)} k is a (locally finite) cover of the plane S X k := {v B(α ), A k (p(v)) chimney} Y k := {v C(α ), γ v (R ) A k (top of chimney) }

156 Higher dimensions In dimension 2: the plane S admits a packing by translates of the base of a chimney. In dimension 3: S will not admit such a packing. Instead we choose A k, k = 1, 2,... isometries such that: A k ( S) = S {A k (chimney base)} k is a (locally finite) cover of the plane S X k := {v B(α ), A k (p(v)) chimney} Y k := {v C(α ), γ v (R ) A k (top of chimney) }. and Y k = A k (X k )

157 ...and Happy Birthday