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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/