Geometry and analysis on hyperbolic manifolds

43
Outline Geometry and analysis on hyperbolic manifolds Yiannis Petridis 1, 2 1 The Graduate Center and Lehman College City University of New York 2 Max-Planck-Institut f ¨ ur Mathematik, Bonn April 20, 2005

Transcript of Geometry and analysis on hyperbolic manifolds

Page 1: Geometry and analysis on hyperbolic manifolds

Outline

Geometry and analysis on hyperbolicmanifolds

Yiannis Petridis1,2

1The Graduate Center and Lehman CollegeCity University of New York

2Max-Planck-Institut fur Mathematik, Bonn

April 20, 2005

Page 2: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Outline

1 Physical background

2 Hyperbolic manifolds

3 Eigenfunctions

4 Periodic orbits

5 Free groups

Page 3: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Systems in physics

Quantum MechanicsFree Particle (non-relativistic) on M satisfies Schrodingerequation

i~∂

∂tΨ(x , t) = − ~2

2m∆Ψ(x , t)

Separate variable Ψ(x , t) = e−iEt/~φ(x). Set ~ = 2m = 1

∆φj + Ejφj = 0, Ej eigenvalues, φj eigenfunctions

Page 4: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Systems in physics

Quantum MechanicsFree Particle (non-relativistic) on M satisfies Schrodingerequation

i~∂

∂tΨ(x , t) = − ~2

2m∆Ψ(x , t)

Separate variable Ψ(x , t) = e−iEt/~φ(x). Set ~ = 2m = 1

∆φj + Ejφj = 0, Ej eigenvalues, φj eigenfunctions

Page 5: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Systems in physics

Quantum MechanicsFree Particle (non-relativistic) on M satisfies Schrodingerequation

i~∂

∂tΨ(x , t) = − ~2

2m∆Ψ(x , t)

Separate variable Ψ(x , t) = e−iEt/~φ(x). Set ~ = 2m = 1

∆φj + Ejφj = 0, Ej eigenvalues, φj eigenfunctions

Page 6: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Statistical Properties of Solutions

Semiclassical limit: Ej →∞

Classically integrable

Barry-Tabor conjecture:Ej independent randomvariables

Localization of φj alongperiodic orbits

Examples

Flat tori, Heisenbergmanifolds

Chaotic systems

Bohigas-Giannoni-Schmit conjecture:Random Matrix Theory

Random WaveConjecture for φj

Examples

Hyperbolic manifolds,Anosov flows

Page 7: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Statistical Properties of Solutions

Semiclassical limit: Ej →∞

Classically integrable

Barry-Tabor conjecture:Ej independent randomvariables

Localization of φj alongperiodic orbits

Examples

Flat tori, Heisenbergmanifolds

Chaotic systems

Bohigas-Giannoni-Schmit conjecture:Random Matrix Theory

Random WaveConjecture for φj

Examples

Hyperbolic manifolds,Anosov flows

Page 8: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Statistical Properties of Solutions

Semiclassical limit: Ej →∞

Classically integrable

Barry-Tabor conjecture:Ej independent randomvariables

Localization of φj alongperiodic orbits

Examples

Flat tori, Heisenbergmanifolds

Chaotic systems

Bohigas-Giannoni-Schmit conjecture:Random Matrix Theory

Random WaveConjecture for φj

Examples

Hyperbolic manifolds,Anosov flows

Page 9: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Statistical Properties of Solutions

Semiclassical limit: Ej →∞

Classically integrable

Barry-Tabor conjecture:Ej independent randomvariables

Localization of φj alongperiodic orbits

Examples

Flat tori, Heisenbergmanifolds

Chaotic systems

Bohigas-Giannoni-Schmit conjecture:Random Matrix Theory

Random WaveConjecture for φj

Examples

Hyperbolic manifolds,Anosov flows

Page 10: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

The Hyperbolic Disc

Model of hyperbolicgeometry

H = {z = x + iy ∈ C, |z| < 1}

Hyperbolic metric

ds2 =dx2 + dy2

(1− (x2 + y2))2

Page 11: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

The Hyperbolic Disc

Model of hyperbolicgeometry

H = {z = x + iy ∈ C, |z| < 1}

Hyperbolic metric

ds2 =dx2 + dy2

(1− (x2 + y2))2

Page 12: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Geodesics in Hyperbolic Disc

Semicircles perpendicular toboundary

Diameters

Page 13: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Geodesics in Hyperbolic Disc

Semicircles perpendicular toboundary

Diameters

Page 14: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

The group SL2(Z)

Upper-half space model

H = {z = x + iy , y > 0}

Identificationsz → z + 1

z → −1z

Group: SL2(Z)

T (z) =az + bcz + d

, ad − bc = 1

a, b, c, d ∈ Z

The fundamental domain ofSL2(Z)

Page 15: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

The group SL2(Z)

Upper-half space model

H = {z = x + iy , y > 0}

Identificationsz → z + 1

z → −1z

Group: SL2(Z)

T (z) =az + bcz + d

, ad − bc = 1

a, b, c, d ∈ Z

The fundamental domain ofSL2(Z)

Page 16: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

The group SL2(Z)

Upper-half space model

H = {z = x + iy , y > 0}

Identificationsz → z + 1

z → −1z

Group: SL2(Z)

T (z) =az + bcz + d

, ad − bc = 1

a, b, c, d ∈ Z

The fundamental domain ofSL2(Z)

Page 17: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

The group SL2(Z)

Upper-half space model

H = {z = x + iy , y > 0}

Identificationsz → z + 1

z → −1z

Group: SL2(Z)

T (z) =az + bcz + d

, ad − bc = 1

a, b, c, d ∈ Z

The fundamental domain ofSL2(Z)

Page 18: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

The group SL2(Z)

Upper-half space model

H = {z = x + iy , y > 0}

Identificationsz → z + 1

z → −1z

Group: SL2(Z)

T (z) =az + bcz + d

, ad − bc = 1

a, b, c, d ∈ Z

The fundamental domain ofSL2(Z)

Page 19: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Arithmetic subgroups of SL 2(Z)

Hecke subgroups Γ0(N)

az + bcz + d

∈ SL2(Z), N|c

Example

Fundamental Domain for Γ0(6)

Page 20: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Tesselations

T� 1F T F

T� 1JF T JF

T� 2UT F UT FU2FT

� 1UT FT� 1U2F

JF

F

0 1� 1

TU2F

1

Figure: Translates of thefundamental domain ofSL2(Z)

Figure: Triangles in thedisc

Page 21: Geometry and analysis on hyperbolic manifolds
Page 22: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Contour plots of eigenfunctions of H/Γ0(7)

Figure: λ = 37.08033 λ = 692.7292

Page 23: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Contour plots of eigenfunctions of H/Γ0(3)

Figure: λ = 26.3467 λ = 60.4397

Page 24: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Distribution of periodic orbits of H/Γ

Periodic orbits are closedgeodesics γ.

Prime Geodesic Theoremπ(x) = {γ, length (γ) ≤ ex}π(x) ∼ x

lnx, x →∞

Prime Number Theoremπ(x) = {p prime, p ≤ x}π(x) ∼ x

lnx, x →∞

Page 25: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Distribution of periodic orbits of H/Γ

Periodic orbits are closedgeodesics γ.

Prime Geodesic Theoremπ(x) = {γ, length (γ) ≤ ex}π(x) ∼ x

lnx, x →∞

Prime Number Theoremπ(x) = {p prime, p ≤ x}π(x) ∼ x

lnx, x →∞

Page 26: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Distribution of periodic orbits of H/Γ

Periodic orbits are closedgeodesics γ.

Prime Geodesic Theoremπ(x) = {γ, length (γ) ≤ ex}π(x) ∼ x

lnx, x →∞

Prime Number Theoremπ(x) = {p prime, p ≤ x}π(x) ∼ x

lnx, x →∞

Page 27: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Free groups

Free group G = F (A1, A2, A3, . . . , Ak )No relations, only AjA

−1j = 1

Cayley graph: tree k = 2

1 Vertices= words2 Edges labelled by

A, B, A−1, B−1

gA↑ A

gB−1 B−1

←−−|g − B−→ gB|↓ A-1

gA−1

Page 28: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Free groups

Free group G = F (A1, A2, A3, . . . , Ak )No relations, only AjA

−1j = 1

Cayley graph: tree k = 2

1 Vertices= words2 Edges labelled by

A, B, A−1, B−1

gA↑ A

gB−1 B−1

←−−|g − B−→ gB|↓ A-1

gA−1

Page 29: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Free groups

Free group G = F (A1, A2, A3, . . . , Ak )No relations, only AjA

−1j = 1

Cayley graph: tree k = 2

1 Vertices= words2 Edges labelled by

A, B, A−1, B−1

gA↑ A

gB−1 B−1

←−−|g − B−→ gB|↓ A-1

gA−1

Page 30: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Free groups

Free group G = F (A1, A2, A3, . . . , Ak )No relations, only AjA

−1j = 1

Cayley graph: tree k = 2

1 Vertices= words2 Edges labelled by

A, B, A−1, B−1

gA↑ A

gB−1 B−1

←−−|g − B−→ gB|↓ A-1

gA−1

Page 31: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Discrete Logarithms

Definitionwl(g) = distance from 1 in the tree

logA(g) = sum of the exponents of A in glogB(g) = sum of the exponents of B in g

Example

logA(B2A3B−2A−1) = 3− 1 = 2wl(B2A3B−2A−1) = 2 + 3 + 2 + 1 = 8

Theorem (Y. Petridis, M. S. Risager 2004)

Gaussian Law for cyclically reduced g

#{g|wl(g) ≤ x ,√

k−1wl(g) logA(g) ∈ [a, b]}

#{g|wl(g) ≤ x}→ 1√

∫ b

ae−u2/2du,

as x →∞

Page 32: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Discrete Logarithms

Definitionwl(g) = distance from 1 in the tree

logA(g) = sum of the exponents of A in glogB(g) = sum of the exponents of B in g

Example

logA(B2A3B−2A−1) = 3− 1 = 2wl(B2A3B−2A−1) = 2 + 3 + 2 + 1 = 8

Theorem (Y. Petridis, M. S. Risager 2004)

Gaussian Law for cyclically reduced g

#{g|wl(g) ≤ x ,√

k−1wl(g) logA(g) ∈ [a, b]}

#{g|wl(g) ≤ x}→ 1√

∫ b

ae−u2/2du,

as x →∞

Page 33: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Discrete Logarithms

Definitionwl(g) = distance from 1 in the tree

logA(g) = sum of the exponents of A in glogB(g) = sum of the exponents of B in g

Example

logA(B2A3B−2A−1) = 3− 1 = 2wl(B2A3B−2A−1) = 2 + 3 + 2 + 1 = 8

Theorem (Y. Petridis, M. S. Risager 2004)

Gaussian Law for cyclically reduced g

#{g|wl(g) ≤ x ,√

k−1wl(g) logA(g) ∈ [a, b]}

#{g|wl(g) ≤ x}→ 1√

∫ b

ae−u2/2du,

as x →∞

Page 34: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Back to H /Γ: Cohomological restrictions

Let α be a differential 1-form with ||α|| = 1.

Theorem (Y. Petridis, M. S. Risager 2004)

Gaussian Law for periodic orbits γ

Let γ have length l(γ). Set [γ, α] =

√vol(M)

2l(γ)

∫γα.

Then, as x →∞,

# {γ ∈ π1(M)|[γ, α] ∈ [a, b], l(γ) ≤ x}#{γ ∈ π1(X )|l(γ) ≤ x}

→ 1√2π

∫ b

ae−u2/2 du

Page 35: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

What are cohomological restrictions

Figure: A surface ofgenus 2

Homology basis A1, A2, A3, A4.

γ =4∑

j=1

njAj

∫γα =

4∑j=1

nj

∫Aj

α

counts (with weights) howmany times γ wraps aroundholes or handles

Page 36: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

What are cohomological restrictions

Figure: A surface ofgenus 2

Homology basis A1, A2, A3, A4.

γ =4∑

j=1

njAj

∫γα =

4∑j=1

nj

∫Aj

α

counts (with weights) howmany times γ wraps aroundholes or handles

Page 37: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

What are cohomological restrictions

Figure: A surface ofgenus 2

Homology basis A1, A2, A3, A4.

γ =4∑

j=1

njAj

∫γα =

4∑j=1

nj

∫Aj

α

counts (with weights) howmany times γ wraps aroundholes or handles

Page 38: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Duality between periods and eigenvalues

Periods Eigenvalues

Page 39: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Duality between periods and eigenvalues

Periods EigenvaluesTrace Formulae

Page 40: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Duality between periods and eigenvalues

Periods EigenvaluesTrace Formulae

Selberg Trace formulaLengths of closedgeodesics

Laplace eigenvalues

Page 41: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Duality between periods and eigenvalues

Periods EigenvaluesTrace Formulae

Selberg Trace formulaLengths of closedgeodesics

Laplace eigenvalues

Ihara Trace formulaLengths of words Eigenvalues ofadjacency matrix

Page 42: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Berry’s Gaussian conjecture

vol(z ∈ A, φj(z) ∈ E)

vol(A)∼ 1√

2πσ

∫E

exp(−u2/2σ) du, j →∞

σ2 =1

vol(H/Γ)

Page 43: Geometry and analysis on hyperbolic manifolds

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Credits for the pictures

1 V. Golovshanski, N. Motrov: preprint, Inst. Appl. Math.Khabarovsk (1982)

2 D. Hejhal, B. Rackner: On the topography of Maasswaveforms for PSL(2, Z ). Experiment. Math. 1 (1992), no.4, 275–305.

3 A. Krieg: http://www.matha.rwth-aachen.de/forschung/fundamentalbereich.html

4 http://mathworld.wolfram.com/5 F. Stromberg:

http://www.math.uu.se/ fredrik/research/gallery/6 H. Verrill: http://www.math.lsu.edu/ verrill/