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

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Outline

1 Physical background

2 Hyperbolic manifolds

3 Eigenfunctions

4 Periodic orbits

5 Free groups

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

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

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

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

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

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

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

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

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

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Geodesics in Hyperbolic Disc

Semicircles perpendicular toboundary

Diameters

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Geodesics in Hyperbolic Disc

Semicircles perpendicular toboundary

Diameters

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)

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)

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)

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)

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)

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)

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

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Contour plots of eigenfunctions of H/Γ0(7)

Figure: λ = 37.08033 λ = 692.7292

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Contour plots of eigenfunctions of H/Γ0(3)

Figure: λ = 26.3467 λ = 60.4397

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 →∞

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 →∞

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 →∞

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

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

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

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

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√

2π

∫ b

ae−u2/2du,

as x →∞

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√

2π

∫ b

ae−u2/2du,

as x →∞

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√

2π

∫ b

ae−u2/2du,

as x →∞

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

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

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

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

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Duality between periods and eigenvalues

Periods Eigenvalues

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Duality between periods and eigenvalues

Periods EigenvaluesTrace Formulae

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups

Duality between periods and eigenvalues

Periods EigenvaluesTrace Formulae

Selberg Trace formulaLengths of closedgeodesics

Laplace eigenvalues

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

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)

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/