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 für Mathematik, Bonn April 20, 2005
Outline 1 Physical background 2 Hyperbolic manifolds 3 Eigenfunctions 4 Periodic orbits 5 Free groups
Systems in physics Quantum Mechanics Free Particle (non-relativistic) on M satisfies Schrödinger equation i 2 Ψ(x, t) = Ψ(x, t) t 2m Separate variable Ψ(x, t) = e iet/ φ(x). Set = 2m = 1 φ j + E j φ j = 0, E j eigenvalues, φ j eigenfunctions
Systems in physics Quantum Mechanics Free Particle (non-relativistic) on M satisfies Schrödinger equation i 2 Ψ(x, t) = Ψ(x, t) t 2m Separate variable Ψ(x, t) = e iet/ φ(x). Set = 2m = 1 φ j + E j φ j = 0, E j eigenvalues, φ j eigenfunctions
Systems in physics Quantum Mechanics Free Particle (non-relativistic) on M satisfies Schrödinger equation i 2 Ψ(x, t) = Ψ(x, t) t 2m Separate variable Ψ(x, t) = e iet/ φ(x). Set = 2m = 1 φ j + E j φ j = 0, E j eigenvalues, φ j eigenfunctions
Statistical Properties of Solutions Semiclassical limit: E j Classically integrable Barry-Tabor conjecture: E j independent random variables Localization of φ j along periodic orbits Examples Flat tori, Heisenberg manifolds Chaotic systems Bohigas-Giannoni- Schmit conjecture: Random Matrix Theory Random Wave Conjecture for φ j Examples Hyperbolic manifolds, Anosov flows
Statistical Properties of Solutions Semiclassical limit: E j Classically integrable Barry-Tabor conjecture: E j independent random variables Localization of φ j along periodic orbits Examples Flat tori, Heisenberg manifolds Chaotic systems Bohigas-Giannoni- Schmit conjecture: Random Matrix Theory Random Wave Conjecture for φ j Examples Hyperbolic manifolds, Anosov flows
Statistical Properties of Solutions Semiclassical limit: E j Classically integrable Barry-Tabor conjecture: E j independent random variables Localization of φ j along periodic orbits Examples Flat tori, Heisenberg manifolds Chaotic systems Bohigas-Giannoni- Schmit conjecture: Random Matrix Theory Random Wave Conjecture for φ j Examples Hyperbolic manifolds, Anosov flows
Statistical Properties of Solutions Semiclassical limit: E j Classically integrable Barry-Tabor conjecture: E j independent random variables Localization of φ j along periodic orbits Examples Flat tori, Heisenberg manifolds Chaotic systems Bohigas-Giannoni- Schmit conjecture: Random Matrix Theory Random Wave Conjecture for φ j Examples Hyperbolic manifolds, Anosov flows
The Hyperbolic Disc Model of hyperbolic geometry H = {z = x + iy C, z < 1} Hyperbolic metric ds 2 = dx 2 + dy 2 (1 (x 2 + y 2 )) 2
The Hyperbolic Disc Model of hyperbolic geometry H = {z = x + iy C, z < 1} Hyperbolic metric ds 2 = dx 2 + dy 2 (1 (x 2 + y 2 )) 2
Geodesics in Hyperbolic Disc Semicircles perpendicular to boundary Diameters
Geodesics in Hyperbolic Disc Semicircles perpendicular to boundary Diameters
The group SL 2 (Z) Upper-half space model H = {z = x + iy, y > 0} The fundamental domain of SL 2 (Z) Identifications z z + 1 z 1 z Group: SL 2 (Z) T (z) = az + b, ad bc = 1 cz + d a, b, c, d Z
The group SL 2 (Z) Upper-half space model H = {z = x + iy, y > 0} The fundamental domain of SL 2 (Z) Identifications z z + 1 z 1 z Group: SL 2 (Z) T (z) = az + b, ad bc = 1 cz + d a, b, c, d Z
The group SL 2 (Z) Upper-half space model H = {z = x + iy, y > 0} The fundamental domain of SL 2 (Z) Identifications z z + 1 z 1 z Group: SL 2 (Z) T (z) = az + b, ad bc = 1 cz + d a, b, c, d Z
The group SL 2 (Z) Upper-half space model H = {z = x + iy, y > 0} The fundamental domain of SL 2 (Z) Identifications z z + 1 z 1 z Group: SL 2 (Z) T (z) = az + b, ad bc = 1 cz + d a, b, c, d Z
The group SL 2 (Z) Upper-half space model H = {z = x + iy, y > 0} The fundamental domain of SL 2 (Z) Identifications z z + 1 z 1 z Group: SL 2 (Z) T (z) = az + b, ad bc = 1 cz + d a, b, c, d Z
Arithmetic subgroups of SL 2 (Z) Example Fundamental Domain for Γ 0 (6) Hecke subgroups Γ 0 (N) az + b cz + d SL 2(Z), N c
Tesselations T 1 F F TF T 1 JF JF T JF T 2 UTF T 1 U 2 F T 1 UTF U 2 F UTF TU 2 F 1 0 1 1 Figure: Translates of the fundamental domain of SL 2 (Z) Figure: Triangles in the disc
Contour plots of eigenfunctions of H/Γ 0 (7) Figure: λ = 37.08033 λ = 692.7292
Contour plots of eigenfunctions of H/Γ 0 (3) Figure: λ = 26.3467 λ = 60.4397
Distribution of periodic orbits of H/Γ Periodic orbits are closed geodesics γ. Prime Geodesic Theorem π(x) = {γ, length (γ) e x } π(x) x lnx, x Prime Number Theorem π(x) = {p prime, p x} π(x) x lnx, x
Distribution of periodic orbits of H/Γ Periodic orbits are closed geodesics γ. Prime Geodesic Theorem π(x) = {γ, length (γ) e x } π(x) x lnx, x Prime Number Theorem π(x) = {p prime, p x} π(x) x lnx, x
Distribution of periodic orbits of H/Γ Periodic orbits are closed geodesics γ. Prime Geodesic Theorem π(x) = {γ, length (γ) e x } π(x) x lnx, x Prime Number Theorem π(x) = {p prime, p x} π(x) x lnx, x
Free groups Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A Free group G = F(A 1, A 2, A 3,..., A k ) No relations, only A j A 1 j = 1 g B gb A -1 ga 1
Free groups Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A Free group G = F(A 1, A 2, A 3,..., A k ) No relations, only A j A 1 j = 1 g B gb A -1 ga 1
Free groups Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A Free group G = F(A 1, A 2, A 3,..., A k ) No relations, only A j A 1 j = 1 g B gb A -1 ga 1
Free groups Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A Free group G = F(A 1, A 2, A 3,..., A k ) No relations, only A j A 1 j = 1 g B gb A -1 ga 1
Discrete Logarithms Definition wl(g) = distance from 1 in the tree log A (g) = sum of the exponents of A in g log B (g) = sum of the exponents of B in g Example log A (B 2 A 3 B 2 A 1 ) = 3 1 = 2 wl(b 2 A 3 B 2 A 1 ) = 2 + 3 + 2 + 1 = 8 Theorem (Y. Petridis, M. S. Risager 2004) Gaussian Law for cyclically reduced g #{g wl(g) x, k 1 wl(g) log A (g) [a, b]} as x #{g wl(g) x} 1 2π b a e u2 /2 du,
Discrete Logarithms Definition wl(g) = distance from 1 in the tree log A (g) = sum of the exponents of A in g log B (g) = sum of the exponents of B in g Example log A (B 2 A 3 B 2 A 1 ) = 3 1 = 2 wl(b 2 A 3 B 2 A 1 ) = 2 + 3 + 2 + 1 = 8 Theorem (Y. Petridis, M. S. Risager 2004) Gaussian Law for cyclically reduced g #{g wl(g) x, k 1 wl(g) log A (g) [a, b]} as x #{g wl(g) x} 1 2π b a e u2 /2 du,
Discrete Logarithms Definition wl(g) = distance from 1 in the tree log A (g) = sum of the exponents of A in g log B (g) = sum of the exponents of B in g Example log A (B 2 A 3 B 2 A 1 ) = 3 1 = 2 wl(b 2 A 3 B 2 A 1 ) = 2 + 3 + 2 + 1 = 8 Theorem (Y. Petridis, M. S. Risager 2004) Gaussian Law for cyclically reduced g #{g wl(g) x, k 1 wl(g) log A (g) [a, b]} as x #{g wl(g) x} 1 2π b a e u2 /2 du,
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] = Then, as x, vol(m) α. 2l(γ) γ # {γ π 1 (M) [γ, α] [a, b], l(γ) x} #{γ π 1 (X) l(γ) x} 1 b e u2 /2 du 2π a
What are cohomological restrictions Figure: A surface of genus 2 Homology basis A 1, A 2, A 3, A 4. 4 γ = n j A j γ j=1 α = 4 n j α A j j=1 counts (with weights) how many times γ wraps around holes or handles
What are cohomological restrictions Figure: A surface of genus 2 Homology basis A 1, A 2, A 3, A 4. 4 γ = n j A j γ j=1 α = 4 n j α A j j=1 counts (with weights) how many times γ wraps around holes or handles
What are cohomological restrictions Figure: A surface of genus 2 Homology basis A 1, A 2, A 3, A 4. 4 γ = n j A j γ j=1 α = 4 n j α A j j=1 counts (with weights) how many times γ wraps around holes or handles
Duality between periods and eigenvalues Periods Eigenvalues
Duality between periods and eigenvalues Periods Trace Formulae Eigenvalues
Duality between periods and eigenvalues Periods Trace Formulae Eigenvalues Lengths of closed geodesics Selberg Trace formula Laplace eigenvalues
Duality between periods and eigenvalues Periods Trace Formulae Eigenvalues Lengths of closed geodesics Lengths of words Selberg Trace formula Ihara Trace formula Laplace eigenvalues Eigenvalues of adjacency matrix
Berry s Gaussian conjecture vol(z A, φ j (z) E) vol(a) 1 2πσ σ 2 = E 1 vol(h/γ) exp( u 2 /2σ) du, j
Credits for the pictures 1 V. Golovshanski, N. Motrov: preprint, Inst. Appl. Math. Khabarovsk (1982) 2 D. Hejhal, B. Rackner: On the topography of Maass waveforms for PSL(2, Z ). Experiment. Math. 1 (1992), no. 4, 275 305. 3 A. Krieg: http://www.matha.rwthaachen.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/