The distribution of prime geodesics for Γ \ H and analogues for free groups

Transcription

1 Outline The distribution of prime geodesics for Γ \ H and analogues for free groups Yiannis Petridis 1 Morten S. Risager 2 1 The Graduate Center and Lehman College City University of New York 2 Aarhus Universitet February 15, 2006

2 Outline 1 Hyperbolic surfaces 2 Closed Geodesics 3 Free groups and discrete logarithms 4 Methods and ideas

3 Discrete groups Upper-half space H = {z = x + iy, y > 0} The fundamental domain of SL 2 (Z) Group: SL 2 (Z) T (z) = az + b, ad bc = 1 cz + d a, b, c, d Z

4 Congruence subgroups of SL 2 (Z) Example Fundamental Domain for Γ 0 (6) Hecke subgroups Γ 0 (N) [ ] a b SL c d 2 (Z), N c

5 Closed geodesics Closed geodesics {γ} = ( {aγa 1, a ) Γ} m 0 γ 0 m 1, m > 1 Lengths of closed geodesic: l(γ) = 2 log m Norm N(γ) = m 2 Hyperbolic metric ds 2 = dx 2 + dy 2 y 2 Example (H/SL 2 (Z)) Q( d), d > 0 ɛ d fundamental unit l(γ) = 2 log ɛ d

6 Distribution of primes Prime number theorem π(x) = {p prime, p x} π(x) x ln x, x Primes in progressions Z Z/nZ π(x, n, a) = {p a mod n, p x} π(x, n, a) 1 x φ(n) ln x Prime geodesic theorem π(x) = {γ, N(γ) x} π(x) x ln x, x Chebotarev density theorem ψ : Γ G finite abelian, β G π(x, β) = {γ, N(γ) x, π(x, β) 1 x G ln x ψ(γ) = β}

7 Examples of the prime geodesic theorem Sarnak: ɛ d x h(d) x 2 2 log x φ : Γ H 1 (H/Γ, Z) = Z 2g = Fix a shifted sublattice β Z 2g /L Get density 1/6, the density of the sublattice.

8 Examples of the prime geodesic theorem Sarnak: ɛ d x h(d) x 2 2 log x φ : Γ H 1 (H/Γ, Z) = Z 2g = Fix a shifted sublattice β Z 2g /L Get density 1/6, the density of the sublattice.

9 Examples of the prime geodesic theorem Sarnak: ɛ d x h(d) x 2 2 log x φ : Γ H 1 (H/Γ, Z) = Z 2g = Fix a shifted sublattice β Z 2g /L Get density 1/6, the density of the sublattice.

10 Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = {γ π(x), φ(γ) = β} π(x, β) (g 1) g x (ln x) g+1 Natural density β A, β T d(a) = lim T β, β T Theorem (P., M. S. Risager 2006) For a set A Z 2g of density d(a) {γ, N(γ) x, φ(γ) A} {γ, N(γ) x} d(a)

11 Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = {γ π(x), φ(γ) = β} π(x, β) (g 1) g x (ln x) g+1 Natural density β A, β T d(a) = lim T β, β T Theorem (P., M. S. Risager 2006) For a set A Z 2g of density d(a) {γ, N(γ) x, φ(γ) A} {γ, N(γ) x} d(a)

12 Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = {γ π(x), φ(γ) = β} π(x, β) (g 1) g x (ln x) g+1 Natural density β A, β T d(a) = lim T β, β T Theorem (P., M. S. Risager 2006) For a set A Z 2g of density d(a) {γ, N(γ) x, φ(γ) A} {γ, N(γ) x} d(a)

13 Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = {γ π(x), φ(γ) = β} π(x, β) (g 1) g x (ln x) g+1 Natural density β A, β T d(a) = lim T β, β T Theorem (P., M. S. Risager 2006) For a set A Z 2g of density d(a) {γ, N(γ) x, φ(γ) A} {γ, N(γ) x} d(a)

14 Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = {γ π(x), φ(γ) = β} π(x, β) (g 1) g x (ln x) g+1 Natural density β A, β T d(a) = lim T β, β T Theorem (P., M. S. Risager 2006) For a set A Z 2g of density d(a) {γ, N(γ) x, φ(γ) A} {γ, N(γ) x} d(a)

15 Theorem (Phillips-Sarnak, Adachi-Sunada ) π(x, β) = {γ π(x), φ(γ) = β} π(x, β) (g 1) g x (ln x) g+1 Natural density β A, β T d(a) = lim T β, β T Theorem (P., M. S. Risager 2006) For a set A Z 2g of density d(a) {γ, N(γ) x, φ(γ) A} {γ, N(γ) x} d(a)

16 Examples of sets and their densities Example Finite sets d(a) = 0 Arithmetic progressions A = {(a 1, a 2,..., a 2g ), a i b i (mod l i )} d(a) = 1 l 1 l 2 l 2g Random sets d(a) = 1/2.

17 Example Points visible from the origin A = {a, gcd(a j ) = 1} d(a) = 1 ζ(2g)

18 Set π A (x) = {γ, N(γ) x, φ(γ) A} Definition We will say that the prime geodesics are equidistributed on a set A Z 2g if π A (x) π(x) d(a), as x, where d(a) is the natural density of A in Z 2g

19 Understanding homology Figure: Genus 2 surface Homology basis A 1, A 2, A 3, A 4. 4 φ(γ) = n j A j j=1 Dual basis ω j with A i ω j = δ ij, n j ω j l(γ)

20 Error terms in prime geodesic theorem π(x) = li(x) + 1/2<s j <1 s 1 j li(x s j ) + R(x) Who? R(x) = Selberg-Randol (1977) O(x 3/4 / log x) Iwaniec (1984) O(x 35/48+ɛ ) Luo-Sarnak (1995) O(x 7/10+ɛ ) Yingchun Cai (2002) O(x 71/102+ɛ ) Conjecture O(x 1/2+ɛ )

21 Error in primes in homology classes π(x, β) = {γ π(x), φ(γ) = β} Phillips-Sarnak π(x, β) = (g 1) g x (ln x) g+1 Sharp: Local Limit Theorem ( 1 + c 1(β) ln x + c ) 2(β) (ln x) 2 + Set σ 2 = (area(γ\h)/2) 1. Then for all β Z 2g we have π(x, β) = ( 1 (2πσ 2 ln x) g exp β, ) ( ) M 1 β x 2σ 2 ln x ln x + o x (ln x) g+1

22 Theorem (P.-Risager 2006) π A (x) = α A α i log x log log x e α,m 1 α /(2σ 2 log x) (2πσ 2 log x) g x log x + o( x log x ). Lemma α A α i m log m e α,m 1 α /(2σ 2 m) (2πσ 2 m) g d(a).

23 Free groups Free group G = F(A 1, A 2, A 3,..., A k ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A g B gb A -1 ga 1

24 Free groups Free group G = F(A 1, A 2, A 3,..., A k ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A g B gb A -1 ga 1

25 Free groups Free group G = F(A 1, A 2, A 3,..., A k ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A g B gb A -1 ga 1

26 Free groups Free group G = F(A 1, A 2, A 3,..., A k ) Cayley graph: tree k = 2 1 Vertices= words 2 Edges labelled by A, B, A 1, B 1 gb 1 B 1 ga A g B gb A -1 ga 1

27 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 ) = = 8 Theorem (P., M. S. Risager 2006) N C (m) = {{g}, wl({g}) m, (log A (g), log B (g)) C} and N(m) = {{g}, wl({g}) m} Then ( 1 NC (m) 2 N(m) + N ) C(m + 1) d(c) N(m + 1)

28 Discrete Logarithms in progressions Let N a1,...,a k (m) = {γ, wl(γ) m, log i (γ) a i (mod l i ) } N(m) = {γ, wl(γ) m} If 2 (l 1, l 2,..., l k ), then N a1,...a k (m) N(m) 1 l 1 l 2 l k, m. If 2 l j, j = 1,..., k, we have m. ( 1 Na1,...,a k (m) + N ) a 1,...,a k (m + 1) 1, 2 N(m) N(m + 1) l 1 l 2 l k

29 Relatively prime discrete logarithms k = 2 N r (m) = {γ, wl(γ) m, gcd(log 1 (γ), log 2 (γ)) = 1} N(m) = {γ, wl(γ) m} Theorem (P., M. S. Risager, 2005) ( 1 Nr (m) 2 N(m) + N ) r (m + 1) 1 N(m + 1) ζ(2)

30 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(log A (g), log B (g)) 1 an intermediate property: do such elements have density d, with 0 < d < 1? Such elements are called test elements.

31 Duality between periods and eigenvalues Periods Eigenvalues

32 Duality between periods and eigenvalues Periods Trace Formulae Eigenvalues

33 Duality between periods and eigenvalues Periods Trace Formulae Eigenvalues Lengths of closed geodesics Selberg Trace formula Laplace eigenvalues

34 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

35 The Selberg zeta function Z (s ) = (1 N(γ 0 ) (s+k) ) {γ 0 } k= s Z (s ) = Tr(R(s ) ) Z

36 The Selberg zeta function Z (s ) = (1 N(γ 0 ) (s+k) ) {γ 0 } k=0 1 Z 1 2s Z (s ) 1 1 2k Z (k ) = Tr(R(s ) R(k ))+ Z

37 The Selberg zeta function Z (s, χ) = (1 χ(γ 0 )N(γ 0 ) (s+k) ) {γ 0 } k= s Z 1 (s, χ) Z 1 2k Z (k, χ) = Tr(R(s, χ) R(k, χ))+ Z

38 The Selberg trace formula allows to estimate R χ (T ) = l(γ) T χ(γ)l(γ) sinh (l(γ)/2). Definition R β (T ) = l(γ) T φ(γ)=β l(γ) sinh (l(γ)/2). Orthogonality of the characters Let χ α ɛ = exp(2πi α, ɛ ). Let χ ɛ (γ) = exp( i j ɛ j R 2g /Z 2g χ ɛ (γ)χ α ɛ dɛ = δ φ(γ)=α γ ω j).

39 Twisted spectral problem h(z) + s 0 (ɛ)(1 s 0 (ɛ))h(z) = 0 h(γ z) = χ ɛ (γ)h(z) Only the neighborhood of χ = 1 is important R β (T ) = 2e T /2 R A (T ) = 2e T /2 B(ε) B(ε) e (s 0(ɛ) 1)T s 0 (ɛ) 1/2 χβ ɛ dɛ + O(e νt ), ν < 1/2 e (s 0(ɛ) 1)T s 0 (ɛ) 1/2 α A α i ct χ α ɛ dɛ + O(T 2g e νt ).

40 Twisted spectral problem h(z) + s 0 (ɛ)(1 s 0 (ɛ))h(z) = 0 h(γ z) = χ ɛ (γ)h(z) Only the neighborhood of χ = 1 is important R β (T ) = 2e T /2 R A (T ) = 2e T /2 B(ε) B(ε) e (s 0(ɛ) 1)T s 0 (ɛ) 1/2 χβ ɛ dɛ + O(e νt ), ν < 1/2 e (s 0(ɛ) 1)T s 0 (ɛ) 1/2 α A α i ct χ α ɛ dɛ + O(T 2g e νt ).

41 The behavior of s 0 (ɛ) Lemma R A (T ) 4e T /2 = s 0 (ɛ) = 1 α A α i T log T s 0 = 1 χ = 1 8π 2 area(γ\h) ɛ, Mɛ + O( ɛ 3 ), M = ω i, ω j exp( α, M 1 α /(2σ 2 T )) (log T )3g/2 (2πσ 2 T ) g +O( T 1/2 )

42 Special features for the graphs If 2 l j, j = 1,..., k, we have ( 1 Na1,...,a k (m) + N ) a 1,...,a k (m + 1) 1, 2 N(m) N(m + 1) l 1 l 2 l k The real character χ 1, χ 2 = 1 contributes

43 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 = H 3, H 3 is the Heisenberg group. (Manin noncommutative modular symbols, K. Chen iterated integrals)