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

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. S. Risager 2006)

NC(m) = |{{g}, wl({g}) ≤ m, (logA(g), logB(g)) ∈ C}| and N(m) = |{{g}, wl({g}) ≤ m}| Then

1 N (m) N (m + 1) C + C → d(C) 2 N(m) N(m + 1) Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Discrete Logarithms in progressions Let

Na1,...,ak (m) = | {γ, wl(γ) ≤ m, logi (γ) ≡ ai (mod li ) } | N(m) = | {γ, wl(γ) ≤ m} |

If 2 6 |(l1, l2,..., lk ), then N (m) 1 a1,...ak → , m → ∞. N(m) l1l2 ··· lk

If 2|lj , j = 1,..., k, we have

1 N (m) N (m + 1) 1 a1,...,ak + a1,...,ak → , 2 N(m) N(m + 1) l1l2 ··· lk m → ∞. Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Relatively prime discrete logarithms

k = 2

Nr (m) = | {γ, wl(γ) ≤ m, gcd(log1(γ), log2(γ)) = 1} | N(m) = | {γ, wl(γ) ≤ m} |

Theorem (P., M. S. Risager, 2005)

1 N (m) N (m + 1) 1 r + r → 2 N(m) N(m + 1) ζ(2) Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Motivation: Asymptotic densities on free groups

Statistical properties of groups elements in a finitely presented groups: e.g. genericity and generic case behavior (Gromov) Question (I.Kapovich, P. Schupp, V. Shpilrain)

On F(A, B) is gcd(logA(g), logB(g)) 6= 1 an intermediate property: do such elements have density d, with 0 < d < 1? Such elements are called test elements. Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Duality between periods and eigenvalues

Periods Eigenvalues Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Duality between periods and eigenvalues

Periods Trace Formulae Eigenvalues Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Duality between periods and eigenvalues

Periods Trace Formulae Eigenvalues

Lengths of closed Selberg Trace formula Laplace eigenvalues geodesics Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Duality between periods and eigenvalues

Periods Trace Formulae Eigenvalues

Lengths of closed Selberg Trace formula Laplace eigenvalues geodesics Lengths of words Ihara Trace formula Eigenvalues of adjacency matrix Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

The Selberg zeta function ∞ Y Y −(s+k) Z (s ) = (1 − N(γ0) )

{γ0} k=0 1 Z 0 (s ) = Tr(R(s )) 1 − 2s Z Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

The Selberg zeta function ∞ Y Y −(s+k) Z (s ) = (1 − N(γ0) )

{γ0} k=0 1 Z 0 1 Z 0 (s ) − (k ) = Tr(R(s ) − R(k ))+ 1 − 2s Z 1 − 2k Z Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

The Selberg zeta function ∞ Y Y −(s+k) Z (s, χ) = (1 − χ(γ0)N(γ0) )

{γ0} k=0 1 Z 0 1 Z 0 (s, χ) − (k, χ) = Tr(R(s, χ) − R(k, χ))+ 1 − 2s Z 1 − 2k Z Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

The Selberg trace formula allows to estimate 0 X χ(γ)l(γ) R (T ) = . χ sinh (l(γ)/2) l(γ)≤T

Definition 0 X l(γ) R (T ) = . β sinh (l(γ)/2) l(γ)≤T φ(γ)=β

Orthogonality of the characters α P R Let χ = exp(2πi hα, i). Let χ(γ) = exp(−i j j γ ωj ). Z α χ(γ)χ d = δφ(γ)=α 2g 2g R /Z Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Twisted spectral problem

∆h(z) + s0()(1 − s0())h(z) = 0

h(γ · z) = χ(γ)h(z)

Only the neighborhood of χ = 1 is important

Z (s0()−1)T T /2 e β νT Rβ(T ) = 2e χ d + O(e ), ν < 1/2 B(ε) s0() − 1/2

Z (s0()−1)T T /2 e X α 2g νT RA(T ) = 2e χ d + O(T e ). s0() − 1/2 B(ε) α∈A |αi |≤cT Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Twisted spectral problem

∆h(z) + s0()(1 − s0())h(z) = 0

h(γ · z) = χ(γ)h(z)

Only the neighborhood of χ = 1 is important

Z (s0()−1)T T /2 e β νT Rβ(T ) = 2e χ d + O(e ), ν < 1/2 B(ε) s0() − 1/2

Z (s0()−1)T T /2 e X α 2g νT RA(T ) = 2e χ d + O(T e ). s0() − 1/2 B(ε) α∈A |αi |≤cT Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

The behavior of s0()

s0 = 1 ⇔ χ = 1 2 8π 3 s0() = 1 − h, Mi + O(|| ), area(Γ\H)

M = hωi , ωj i

Lemma

R (T ) X exp(−hα, M−1αi/(2σ2T )) (log T )3g/2 A = +O( ) 4eT /2 (2πσ2T )g T 1/2 α√∈A |αi |≤ T log T Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Special features for the graphs

If 2|lj , j = 1,..., k, we have

1 N (m) N (m + 1) 1 a1,...,ak + a1,...,ak → , 2 N(m) N(m + 1) l1l2 ··· lk

The real character χ 6= 1, χ2 = 1 contributes Hyperbolic surfaces Closed Geodesics Free groups and discrete logarithms Methods and ideas

Conclusions Equidistribution of closed geodesics in general sets of homology Similar results for the distribution of discrete logarithms in free groups Applications to infinite group theory

Open problems Number theoretic interpretation of the results on homology classes

Noncommutative analogues: Γ → N, e.g. N = H3, H3 is the Heisenberg group. (Manin noncommutative modular symbols, K. Chen iterated integrals)