Outline The distribution of prime geodesics for Γ \ H and analogues for free groups Yiannis Petridis1 Morten S. Risager2 1The Graduate Center and Lehman College City University of New York 2Aarhus Universitet February 15, 2006 Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Outline 1 Hyperbolic surfaces 2 Closed Geodesics 3 Free groups and discrete logarithms 4 Methods and ideas Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Discrete groups The fundamental domain of SL2(Z) Upper-half space H = {z = x + iy, y > 0} Group: SL2(Z) az + b T (z) = , ad − bc = 1 cz + d a, b, c, d ∈ Z Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Congruence subgroups of SL2(Z) Example Fundamental Domain for Γ0(6) Hecke subgroups Γ0(N) a b ∈ SL ( ), N|c c d 2 Z Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Closed geodesics Closed geodesics {γ} = {aγa−1, a ∈ Γ} m 0 γ ∼ , m > 1 0 m−1 Lengths of closed geodesic: l(γ) = 2 log m Norm N(γ) = m2 Example (H/SL2(Z)) √ Hyperbolic metric Q( d), d > 0 fundamental unit dx2 + dy 2 d ds2 = y 2 l(γ) = 2 log d Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Distribution of primes Prime number theorem Prime geodesic theorem π(x) = |{p prime, p ≤ x}| π(x) = |{γ, N(γ) ≤ x}| x x π(x) ∼ , x → ∞ π(x) ∼ , x → ∞ ln x ln x Primes in progressions Chebotarev density theorem Z → Z/nZ ψ :Γ → G finite abelian, β ∈ G π(x, n, a) = |{p ≡ a mod n, p ≤ x}| π(x, β) = |{γ, N(γ) ≤ x, 1 x π(x, n, a) ∼ ψ(γ) = β}| φ(n) ln x 1 x π(x, β) ∼ |G| ln x Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Examples of the prime geodesic theorem X x2 Sarnak: h(d) ∼ 2 log x d ≤x ∼ 2g ∼ φ :Γ → H1(H/Γ, Z) = Z = Fix a shifted sublattice β ∈ Z2g/L Get density 1/6, the density of the sublattice. Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Examples of the prime geodesic theorem X x2 Sarnak: h(d) ∼ 2 log x d ≤x ∼ 2g ∼ φ :Γ → H1(H/Γ, Z) = Z = Fix a shifted sublattice β ∈ Z2g/L Get density 1/6, the density of the sublattice. Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Examples of the prime geodesic theorem X x2 Sarnak: h(d) ∼ 2 log x d ≤x ∼ 2g ∼ φ :Γ → H1(H/Γ, Z) = Z = Fix a shifted sublattice β ∈ Z2g/L Get density 1/6, the density of the sublattice. Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = |{γ ∈ π(x), φ(γ) = β}| x π(x, β) ∼ (g − 1)g (ln x)g+1 Natural density |β ∈ A, kβk ≤ T | d(A) = lim T →∞ |β, kβk ≤ T | Theorem (P., M. S. Risager 2006) For a set A ⊂ Z2g of density d(A) |{γ, N(γ) ≤ x, φ(γ) ∈ A}| → d(A) |{γ, N(γ) ≤ x}| Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = |{γ ∈ π(x), φ(γ) = β}| x π(x, β) ∼ (g − 1)g (ln x)g+1 Natural density |β ∈ A, kβk ≤ T | d(A) = lim T →∞ |β, kβk ≤ T | Theorem (P., M. S. Risager 2006) For a set A ⊂ Z2g of density d(A) |{γ, N(γ) ≤ x, φ(γ) ∈ A}| → d(A) |{γ, N(γ) ≤ x}| Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = |{γ ∈ π(x), φ(γ) = β}| x π(x, β) ∼ (g − 1)g (ln x)g+1 Natural density |β ∈ A, kβk ≤ T | d(A) = lim T →∞ |β, kβk ≤ T | Theorem (P., M. S. Risager 2006) For a set A ⊂ Z2g of density d(A) |{γ, N(γ) ≤ x, φ(γ) ∈ A}| → d(A) |{γ, N(γ) ≤ x}| Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = |{γ ∈ π(x), φ(γ) = β}| x π(x, β) ∼ (g − 1)g (ln x)g+1 Natural density |β ∈ A, kβk ≤ T | d(A) = lim T →∞ |β, kβk ≤ T | Theorem (P., M. S. Risager 2006) For a set A ⊂ Z2g of density d(A) |{γ, N(γ) ≤ x, φ(γ) ∈ A}| → d(A) |{γ, N(γ) ≤ x}| Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = |{γ ∈ π(x), φ(γ) = β}| x π(x, β) ∼ (g − 1)g (ln x)g+1 Natural density |β ∈ A, kβk ≤ T | d(A) = lim T →∞ |β, kβk ≤ T | Theorem (P., M. S. Risager 2006) For a set A ⊂ Z2g of density d(A) |{γ, N(γ) ≤ x, φ(γ) ∈ A}| → d(A) |{γ, N(γ) ≤ x}| Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = |{γ ∈ π(x), φ(γ) = β}| x π(x, β) ∼ (g − 1)g (ln x)g+1 Natural density |β ∈ A, kβk ≤ T | d(A) = lim T →∞ |β, kβk ≤ T | Theorem (P., M. S. Risager 2006) For a set A ⊂ Z2g of density d(A) |{γ, N(γ) ≤ x, φ(γ) ∈ A}| → d(A) |{γ, N(γ) ≤ x}| Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Examples of sets and their densities Example Finite sets d(A) = 0 Arithmetic progressions A = {(a1, a2,..., a2g), ai ≡ bi (mod li )} 1 d(A) = l1l2 ··· l2g Random sets d(A) = 1/2. Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Example Points visible from the origin A = {a, gcd(aj ) = 1} 1 d(A) = ζ(2g) Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Set πA(x) = |{γ, N(γ) ≤ x, φ(γ) ∈ A}| Definition We will say that the prime geodesics are equidistributed on a set A ⊆ Z2g if π (x) A → d(A), as x → ∞, π(x) where d(A) is the natural density of A in Z2g Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Understanding homology Homology basis A1, A2, A3, A4. 4 X φ(γ) = nj Aj j=1 R Dual basis ωj with ωj = δij , Ai |nj | ≤ ||ωj ||∞l(γ) Figure: Genus 2 surface Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Error terms in prime geodesic theorem X −1 sj π(x) = li(x) + sj li(x ) + R(x) 1/2<sj <1 Who? R(x) = Selberg-Randol (1977) O(x3/4/ log x) Iwaniec (1984) O(x35/48+) Luo-Sarnak (1995) O(x7/10+) Yingchun Cai (2002) O(x71/102+) Conjecture O(x1/2+) Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Error in primes in homology classes π(x, β) = |{γ ∈ π(x), φ(γ) = β}| Phillips-Sarnak x c (β) c (β) π(x, β) = (g − 1)g 1 + 1 + 2 + ··· (ln x)g+1 ln x (ln x)2 Sharp: Local Limit Theorem − Set σ2 = (area(Γ\H)/2) 1. Then for all β ∈ Z2g we have 1 hβ, M−1βi x x π(x, β) = exp − + o (2πσ2 ln x)g 2σ2 ln x ln x (ln x)g+1 Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Theorem (P.-Risager 2006) −1 2 X e−hα,M αi/(2σ log x) x x π (x) = + o( ). A (2πσ2 log x)g log x log x √ α∈A |αi |≤ log x log log x Lemma −1 2 X e−hα,M αi/(2σ m) → d(A). (2πσ2m)g α√∈A |αi |≤ m log m Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Free groups Free group G = F(A1, A2, A3,..., Ak ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A−1, B−1 gA ↑ A −1 | gB−1 ←−−B g −−→B gB | ↓ A-1 gA−1 Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Free groups Free group G = F(A1, A2, A3,..., Ak ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A−1, B−1 gA ↑ A −1 | gB−1 ←−−B g −−→B gB | ↓ A-1 gA−1 Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Free groups Free group G = F(A1, A2, A3,..., Ak ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A−1, B−1 gA ↑ A −1 | gB−1 ←−−B g −−→B gB | ↓ A-1 gA−1 Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Free groups Free group G = F(A1, A2, A3,..., Ak ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A−1, B−1 gA ↑ A −1 | gB−1 ←−−B g −−→B gB | ↓ A-1 gA−1 Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas Discrete Logarithms Definition wl(g) = distance from 1 in the tree logA(g) = sum of the exponents of A in g logB(g) = sum of the exponents of B in g Example 2 3 −2 −1 logA(B A B A ) = 3 − 1 = 2 wl(B2A3B−2A−1) = 2 + 3 + 2 + 1 = 8 Theorem (P., M.