Geometric Identities RIMS Seminar 2012 Greg McShane June 5, 2012 - PowerPoint PPT Presentation

Geometric Identities RIMS Seminar 2012 Greg McShane June 5, 2012

Part I Introduction

Surfaces ◮ Σ is a surface ◮ totally geodesic boundary ◮ finite volume hyperbolic structure

Surfaces ◮ Σ is a surface ◮ totally geodesic boundary ◮ finite volume hyperbolic structure ◮ Γ ≃ π 1 (Σ) ◮ Λ = limit set of Γ ◮ CC (Λ) / Γ = Σ

Spectra definition length | tr γ | = 2 cosh( 1 Closed geodesic [ γ ] , γ � = 1 ∈ Γ 2 ℓ ( γ ))

Spectra definition length | tr γ | = 2 cosh( 1 Closed geodesic [ γ ] , γ � = 1 ∈ Γ 2 ℓ ( γ )) Simple closed geodesic same as above + no self intersection same as above

Spectra definition length | tr γ | = 2 cosh( 1 Closed geodesic [ γ ] , γ � = 1 ∈ Γ 2 ℓ ( γ )) Simple closed geodesic same as above + no self intersection same as above γ ∗ shortest arc Ortho geodesic joins 2 geodesic see below tanh 2 is cross ratio boundary components

Spectra definition length | tr γ | = 2 cosh( 1 Closed geodesic [ γ ] , γ � = 1 ∈ Γ 2 ℓ ( γ )) Simple closed geodesic same as above + no self intersection same as above γ ∗ shortest arc Ortho geodesic joins 2 geodesic see below tanh 2 is cross ratio boundary components γ.β.α = 1 ∈ Γ Immersed pair of pants ( ℓ ( α ) , ℓ ( β ) , ℓ ( γ ))

Spectra definition length | tr γ | = 2 cosh( 1 Closed geodesic [ γ ] , γ � = 1 ∈ Γ 2 ℓ ( γ )) Simple closed geodesic same as above + no self intersection same as above γ ∗ shortest arc Ortho geodesic joins 2 geodesic see below tanh 2 is cross ratio boundary components γ.β.α = 1 ∈ Γ Immersed pair of pants ( ℓ ( α ) , ℓ ( β ) , ℓ ( γ )) Embedded pair of pants same as above but [ γ ] , [ β ] , [ α ] simple,disjoint same as above

Spectra definition length | tr γ | = 2 cosh( 1 Closed geodesic [ γ ] , γ � = 1 ∈ Γ 2 ℓ ( γ )) Simple closed geodesic same as above + no self intersection same as above γ ∗ shortest arc Ortho geodesic joins 2 geodesic see below tanh 2 is cross ratio boundary components γ.β.α = 1 ∈ Γ Immersed pair of pants ( ℓ ( α ) , ℓ ( β ) , ℓ ( γ )) Embedded pair of pants same as above but [ γ ] , [ β ] , [ α ] simple,disjoint same as above Ortho geodesic is a pair α, β ∈ Γ , [ α ] , [ β ] ⊂ ∂ Σ ( α − − β − )( α + − β + ) ( α − − β + )( α + − β − ) = tanh 2 (1 2 ℓ ( γ ∗ ))

Spectra ◮ Length spectrum = { lengths of closed geodesics }

Spectra ◮ Length spectrum = { lengths of closed geodesics } ◮ Simple length spectrum = { lengths of simple closed geods }

Spectra ◮ Length spectrum = { lengths of closed geodesics } ◮ Simple length spectrum = { lengths of simple closed geods } ◮ Ortho spectrum = { lengths of ortho geodesics }

Spectra ◮ Length spectrum = { lengths of closed geodesics } ◮ Simple length spectrum = { lengths of simple closed geods } ◮ Ortho spectrum = { lengths of ortho geodesics } ◮ Pant’s spectrum = { lengths of embedded pants }

Spectra ◮ Length spectrum = { lengths of closed geodesics } ◮ Simple length spectrum = { lengths of simple closed geods } ◮ Ortho spectrum = { lengths of ortho geodesics } ◮ Pant’s spectrum = { lengths of embedded pants } ◮ δ = Hausdorff dimension of the limit set. ◮ Vol(Σ) ◮ Vol( ∂ Σ)

Length spectrum ◮ (Weyl) Spectrum of Laplacian determines the area N Γ ( t ) := |{ eigenvalues of∆ H / Γ < t }| ∼ Vol( H / Γ) t 4 π

Length spectrum ◮ (Weyl) Spectrum of Laplacian determines the area N Γ ( t ) := |{ eigenvalues of∆ H / Γ < t }| ∼ Vol( H / Γ) t 4 π ◮ (Huber, Selberg) Length spectrum determines the spectrum of the Laplacian.

Length spectrum ◮ (Weyl) Spectrum of Laplacian determines the area N Γ ( t ) := |{ eigenvalues of∆ H / Γ < t }| ∼ Vol( H / Γ) t 4 π ◮ (Huber, Selberg) Length spectrum determines the spectrum of the Laplacian. ◮ (Margulis/Sullivan) Length spectrum determines the Hausdorff dimension N Γ ( t ) := |{ primitive geodesics ℓ ( α ) < t }| ∼ e δ t δ t .

Length spectrum ◮ (Weyl) Spectrum of Laplacian determines the area N Γ ( t ) := |{ eigenvalues of∆ H / Γ < t }| ∼ Vol( H / Γ) t 4 π ◮ (Huber, Selberg) Length spectrum determines the spectrum of the Laplacian. ◮ (Margulis/Sullivan) Length spectrum determines the Hausdorff dimension N Γ ( t ) := |{ primitive geodesics ℓ ( α ) < t }| ∼ e δ t δ t . ◮ (Wolpert) Length spectrum determines the isometry type of the surface up to finitely many choices

Trace formula ◮ h even function, satisfying a growth condition ◮ ˆ h Fourier transform

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

Simple Length Spectra ◮ (Wolpert) Simple length spectrum determines the surface up to finitely many choices. ◮ (Mirkzahani) N ( t ) := |{ simple geodesics ℓ ( α ) < t }| ∼ C ( H / Γ) t 6 g − 6 . g = genus of Σ.

Part II Identities

Basmajian Identity Theorem (1992) � � 1 � 2 sinh − 1 = ℓ ( δ ) sinh( ℓ ( α ∗ ) α ∗

Basmajian Identity Theorem (1992) � � 1 � 2 sinh − 1 = ℓ ( δ ) sinh( ℓ ( α ∗ ) α ∗ � � 1 �� � Ball radius = sinh − 1 Vol n − 1 = Vol n − 1 ( ∂ M ) sinh( ℓ ( α ∗ ) α ∗

Bridgeman-Kahn Identity Theorem (2008) � 1 � � 2 π Vol ( M ) = 8 L cosh 2 ( ℓ ( α ∗ ) / 2) α ∗

Bridgeman-Kahn Identity Theorem (2008) � 1 � � 2 π Vol ( M ) = 8 L cosh 2 ( ℓ ( α ∗ ) / 2) α ∗ ◮ Dilogarithm � z � z k log(1 − x ) Li 2 ( z ) = k 2 = − dx x 0

Bridgeman-Kahn Identity Theorem (2008) � 1 � � 2 π Vol ( M ) = 8 L cosh 2 ( ℓ ( α ∗ ) / 2) α ∗ ◮ Dilogarithm � z � z k log(1 − x ) Li 2 ( z ) = k 2 = − dx x 0 ◮ Roger’s dilogarithm L ( x ) = Li 2 ( x ) + 1 2 log | x | log(1 − x ) , x < 1 . L ′ ( x ) = 1 � log(1 − x ) + log( x ) � . 1 − x 2 x

Bridgeman-Kahn Identity in general Theorem � 1 � � 2 π Vol ( M ) = 8 L cosh 2 ( ℓ ( α ∗ ) / 2) α ∗

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

Bridgeman-Kahn Identity in general Theorem � 1 � � 2 π Vol ( M ) = 8 L cosh 2 ( ℓ ( α ∗ ) / 2) α ∗ Exist F n such that for any hyperbolic n-manifold M with totally geodesic boundary � F n ( ℓ ( α ∗ )) Vol( M ) = β 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.

Identity for embedded pants Σ has a single boundary component of length ℓ ( δ ) ≥ 0 ◮ Punctured torus ℓ ( δ ) = 0 1 + e ℓ ( α ) = 1 1 � 2 α

Identity for embedded pants Σ has a single boundary component of length ℓ ( δ ) ≥ 0 ◮ Punctured torus ℓ ( δ ) = 0 1 + e ℓ ( α ) = 1 1 � 2 α ◮ One-holed torus 1 � � 2 ( ℓ ( α ) − ℓ ( δ )) 1 + e � log = ℓ ( δ ) 1 2 ( ℓ ( α )+ ℓ ( δ )) 1 + e α

Identity for embedded pants Σ has a single boundary component of length ℓ ( δ ) ≥ 0 ◮ Punctured torus ℓ ( δ ) = 0 1 + e ℓ ( α ) = 1 1 � 2 α ◮ One-holed torus 1 � � 2 ( ℓ ( α ) − ℓ ( δ )) 1 + e � log = ℓ ( δ ) 1 2 ( ℓ ( α )+ ℓ ( δ )) 1 + e α ◮ One-holed genus g � 1 � 2 ( ℓ ( α )+ ℓ ( β ) − ℓ ( δ )) 1 + e � log = ℓ ( δ ) 1 2 ( ℓ ( α )+ ℓ ( β )+ ℓ ( δ )) 1 + e P P is an embedded pair of pants with waist δ and legs α, β

Identity for embedded pants Σ has a single boundary component of length ℓ ( δ ) ≥ 0 ◮ Punctured torus ℓ ( δ ) = 0 1 + e ℓ ( α ) = 1 1 � 2 α ◮ One-holed torus 1 � � 2 ( ℓ ( α ) − ℓ ( δ )) 1 + e � log = ℓ ( δ ) 1 2 ( ℓ ( α )+ ℓ ( δ )) 1 + e α ◮ One-holed genus g � 1 � 2 ( ℓ ( α )+ ℓ ( β ) − ℓ ( δ )) 1 + e � log = ℓ ( δ ) 1 2 ( ℓ ( α )+ ℓ ( β )+ ℓ ( δ )) 1 + e P P is an embedded pair of pants with waist δ and legs α, β P on a holed torus is pants with waist δ and legs α, α

Luo-Tan Theorem (2010) � � f ( P ) + g ( T ) = 2 π Vol ( M ) P T where

Part III Proofs

Decompositions Given an identity : what is the associated decomposition of the surface ?

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

Limit set Λ:= limit set. Theorem (Ahlfors) M = H / Γ is geometrically finite, and Λ c � = ∅ then Λ has measure zero.