SECTOR ESTIMATES FOR HYPERBOLIC ISOMETRIES 0 JEAN BOURGAIN, ALEX KONTOROVICH, AND PETER SARNAK 1 0 2 Abstract. WeprovevariousorbitalcountingstatementsforFuch- n sian groups of the second kind. These are of independent interest, a J andalsoareusedinthecompanionpaper[BK09]toproduceprimes 5 in the Affine Linear Sieve. 2 ] T N Contents . h t 1. Introduction 1 a m 2. Preliminaries 6 [ 3. Proof of Theorem 1.5 19 1 4. Proof of Theorem 1.13 23 v 1 5. Proof of Theorem 1.14 28 4 5 6. Proof of Theorem 1.15 29 4 . References 31 1 0 0 1 v: 1. Introduction i X Let Γ < G = PSL(2,R) be a non-elementary, geometrically finite r a group of isometries of the upper half plane H. In the case when Γ has fundamental domain with finite hyperbolic area, much effort has gone into understanding the asymptotic behavior of the number of points in a Γ-orbit which lie in an expanding region inside G, e.g. [Sel65, LP82, Goo83a, DRS93, EM93, Kon09, KO09]. Such regions can be decomposed into their harmonics, and hence one can recover many counting statements from, say, taking the Cartan decomposition Date: January 25, 2010. Bourgain is partially supported by NSF grant DMS-0808042. Kontorovich is partially supported by NSF grants DMS-0802998 and DMS- 0635607,and the Ellentuck Fund at IAS. Sarnak is partially supported by NSF grant DMS-0758299. 1 2 JEAN BOURGAIN,ALEX KONTOROVICH,ANDPETER SARNAK G = KA+K and studying the set of γ Γ in a ball of expanding radius ∈ with a given harmonic on right and left K-types. Here cosθ sinθ K = SO(2) = k = : θ [0,2π) θ sinθ cosθ ∈ (cid:26) (cid:18) − (cid:19) (cid:27) and e−t/2 0 A+ = a = ,t 0 . t 0 et/2 ≥ (cid:26) (cid:18) (cid:19) (cid:27) Thestateoftheartinthisdirectioninthefinite-volumecaseisdueto Good [Goo83a]. Let 0 θ (g),θ (g) < π and t(g) > 0 be the functions 1 2 ≤ in the Cartan decomposition of G, so that g = k a k .1 ± θ1(g) t(g) θ2(g) Theorem 1.1 ([Goo83a], Thm 4). Let Γ < G be a lattice and fix integers n and k. Let 0 = λ < λ λ < 1/4 0 1 N ≤ ··· ≤ be the eigenvalues of the Laplacian on Γ H below 1/4. Write \ λ = s (1 s ) (1.2) j j j − with s > 1/2. Then there are constants c ,...,c C depending on j 0 N ∈ n and k such that N e2inθ1(γ) e2ikθ2(γ) = 1 1 c T2+ c (n,k)T2sj+O (T4/3),2 n=0 k=0 0 j n,k γ∈Γ j=1 kXγk0. Let θ (g) be defined as 1arg g·i−i k k 1 2 g·i+i (thatis,mapg totheunitdiskD; theargumentoftheimagedeterminesθ1)(cid:16). The(cid:17)n a−1 k−1 g is in K, whence θ is determined. t(g) θ1(g) 2 2 The exponent 4/3 has never been improved, even for any specific Γ. SECTOR ESTIMATES FOR HYPERBOLIC ISOMETRIES 3 below 1/4 [Pat75]. In fact, the spectrum contains no point eigenvalues at all unless δ > 1/2, in which case the base eigenvalue is [Pat76] λ = δ(1 δ). 0 − Corresponding to λ is the base eigenfunction ϕ , which can be real- 0 0 ized explicitly as the integral of a Poisson kernel against the so-called Patterson-Sullivan measure µ [Pat76, Sul84], supported on the limit set Λ ∂H. Roughly speaking, µ is the weak∗ limit as s δ+ of the ⊂ → measures exp( sd(o,γ o))1 µ (x) := γ∈Γ − · x=γo. (1.4) s exp( sd(o,γ o)) P γ∈Γ − · Here d( , ) is the hyperbolic distance, and o is the origin (or any base P point) in· ·D. Let µ denote the Fourier coefficients of the measure µ. Our first result is b Theorem 1.5. Let Γ be a Fuchsian group of the second kind with critical exponent δ > 1/2. Let 0 < δ(1 δ) = λ < λ λ < 1/4 (1.6) 0 1 N − ≤ ··· ≤ be the eigenvalues of the Laplacian on Γ H below 1/4, and use the nota- \ tion (1.2). Then for integers n and k, there are constants c ,...,c 1 N C depending on n and k such that ∈ N Γ(δ 1/2) e2inθ1(γ) e2ikθ2(γ) = µ(2n)µ(2k) π1/2 − T2δ + c (n,k)T2sj j Γ(δ +1) γ∈Γ j=1 kXγk 5/6. Then there exists some ramification number B depending on Γ such that Θ = 5/6 is a spectral gap for Γ. We also require the following Sobolev-type norm. Fix T 1 and let ≥ X ,X ,X be a basis for the Lie algebra g, cf. (2.5). Then define 1 2 3 { } the norm by ∞,T S f = max sup dπ(X).f(g) , ∞,T S X∈{0,X1,X2,X3}g∈G,kgk 1/2, ramification number B, and | | ≤ spectral gap Θ. Then for any γ Γ and integer q 1 with q = q′q′′ 0 ∈ ≥ and q′ B, | 1 f(γ) = f(γ)+  q′ [Γ : Γ(q)] E γ∈kXγγ0k<·ΓT(q) kXγγk∈<ΓT    +O (1+ ∞,Tf)6/7T76·2δ+17·2Θ . S (cid:18) (cid:19) Here q′ T2δ−α0, with α0 > 0, and all implied constants are indepen- E ≪ dent of q′′ and γ . 0 The application in the companion paper [BK09] requires the following two estimates, which we derive as consequences of the above. Theorem 1.14. Assume Γ has critical exponent δ > 1/2. Let v and w be vectors in Z2, and n Z. Let 0 < K < T < N be parameters ∈ with K and N/K (and a fortiori N/T ). Assume → ∞ → ∞ → ∞ that N < n < N, w < N, v 1, and n < v w T. Then K | | | | T | | ≤ | | | || | N T2δ 1 vγ,w n < +O T34+2δ41(logT)1/4 . |h i− | K ≫ K kXγγk∈<ΓT (cid:26) (cid:27) (cid:16) (cid:17) 6 JEAN BOURGAIN,ALEX KONTOROVICH,ANDPETER SARNAK Theorem 1.15. Assume Γ has δ > 1/2 and spectral gap Θ. Let N,K, and T be as above, and fix q 1. Fix y = (y ,y ) and (c,d) in Z2 such 1 2 ≥ that y < N, (c,d) < N and y < T (c,d) . Then | | | | T | | | | N T2δ 1 (c,d)γ y < 1 (c,d)γ y(modq) +T672δ+172Θ, | − | K ≡ ≪ K1+δq2 γ∈Γ (cid:26) (cid:27) (cid:26) (cid:27) kXγk 1/2, use the notation (1.6) and (1.2), and let ϕ be a Laplace eigenfunction corresponding to λ = s (1 s ). The j j j j − decomposition into irreducibles of the right regular representation on the vector space V = L2(Γ G) is of the form [GGPS66] \ V = V V V V , (2.1) ϕ0 ⊕ ϕ1 ⊕···⊕ ϕN ⊕ temp where each V is the G-span of the eigenfunction ϕ , and is isomorphic ϕj j as a G-representation to the complementary series representation with parameter s (in our normalization, the principal series representations j lie on the critical line Re(s) = 1/2). The reducible space V consists temp of the tempered spectrum. It will be convenient to use both the automorphic model above and, say, the line model, which we recall now. Fix s > 1/2 and let (π,V ) s denote the line model for the complementary series representation with parameter s > 1/2 [GGPS66]. That is, let G act on functions f : R C with action given by → a b ax c π .f(x) = bx+d −2sf − . c d |− | bx+d (cid:18) (cid:19) (cid:18)− (cid:19) The intertwining operator : V V is defined by s 1−s I → f(x) .f(y) := dx. (2.2) I x y 2(1−s) R Z | − | SECTOR ESTIMATES FOR HYPERBOLIC ISOMETRIES 7 For f ,f V , the pairing is 1 2 s ∈ f ,f = f (x) .f (x)dx. (2.3) 1 2 1 2 h i I R Z Then V consists of functions f with f,f < . s h i ∞ 2.2. Raising, Lowering, and Casimir Operators. We return to the automorphic model and let be one such irre- H ducible V . The dense subspace ∞ of smooth vectors is infinite ϕj H dimensional, and decomposes further into one-dimensional K-isotypic components: ∞ = (2k), (2.4) H H k∈Z M each (2k) = C v consisting of functions of “weight 2k”, that is, 2k H · those functions v ∞ which transform as: 2k ∈ H v (gk ) = e2ikθv (g). 2k θ 2k Let 1 0 0 1 0 0 h = , e = , and f = (2.5) 0 1 0 0 1 0 (cid:18) − (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) denote a basis for the Lie algebra g = sl(2,R). The ladder (raising and lowering) operators and in the complexified Lie algebra gC R L are defined by: = h+i(e+f), = h i(e+f). R L − Recall also the Casimir element , which generates the center of the C universal enveloping algebra (gC), and acts on as scalar multipli- U H cation by 2λ = 2s(1 s): − − − 1 = h2 +ef +fe. C 2 We will require expressions for these operators in Cartan coordinates (θ ,t,θ ), corresponding to g = k a k . For the Casimir operator, 1 2 θ1 t θ2 these can be found in many places, e.g. [Kna86, p. 216 and p. 700], [CM82, p. 884], or [KT00, 7]: § 1 ∂2 ∂ ∂2 ∂2 2cosht ∂2 = +(cotht) +csch2t + . (2.6) 2 C ∂t2 ∂t ∂θ2 ∂θ2 − sinh2t ∂θ ∂θ (cid:18) 1 2(cid:19) 1 2 The raising and lowering operators are not as readily available in Cartan coordinates in the literature, so we derive their expression here. 8 JEAN BOURGAIN,ALEX KONTOROVICH,ANDPETER SARNAK Lemma 2.7. InKA+K coordinates, the raising and loweringoperators are: ∂ ∂ ∂ = e2iθ2 icsch(t) +2 +icoth(t) , R − ∂θ ∂t ∂θ (cid:18) 1 2(cid:19) and ∂ ∂ ∂ = e−2iθ2 icsch(t) +2 icoth(t) . L ∂θ ∂t − ∂θ (cid:18) 1 2(cid:19) Proof. Set V = e + f and Y = e f, so that e = 1V + 1Y and − 2 2 f = 1V 1Y. Let betheCartandecomposition, thatis, theinjection 2 −2 J : K A+ K G. J × × → For a point x = (θ ,t,θ ), we compute the derivation 1 2 D( ) : T T = g, x g J → as follows. Any element T is of the form x T = ξ Y +ηh+ξ Y , x 1 1 2 2 where Y = Y = Y k = so(2), but each is interpreted as a vector 1 2 ∈ field in its respective component. We compute keeping only first order terms: (k (I +ξ Y ), a (I +ηh), k (I +ξ Y )) 1 1 1 t 2 2 2 J = k (I +ξ Y )a (I +ηh)k (I +ξ Y ) 1 1 1 t 2 2 2 = k a k (k−1a−1(I +ξ Y )a k )(k−1(I +ηh)k )(I +ξ Y ) 1 t 2 2 t 1 1 t 2 2 2 2 2 = k a k (I +ad(k−1)ad(a−1)ξ Y )(I +ad(k−1)ηh)(I +ξ Y ) 1 t 2 2 t 1 1 2 2 2 k a k (I +ad(k−1)ad(a−1)ξ Y +ad(k−1)ηh+ξ Y ) ≈ 1 t 2 2 t 1 1 2 2 2 = k a k I +ad(k−1) ad(a−1)ξ Y +ηh+ad(k )ξ Y . 1 t 2 2 t 1 1 2 2 2 ! (cid:26) (cid:27) Hence D( ) : ξ Y +ηh+ξ Y ad(k−1) ad(a−1)ξ Y +ηh+ad(k )ξ Y . J 1 1 2 2 7→ 2 t 1 1 2 2 2 (cid:26) (cid:27) SECTOR ESTIMATES FOR HYPERBOLIC ISOMETRIES 9 We compute 1 1 1 1 ad(a−1)Y = a−1Ya = e−te etf = e−t( V + Y) et( V Y) t t t − 2 2 − 2 − 2 = cosh(t)Y sinh(t)V, − ad(k−1)Y = Y, ad(k−1)V = cos(2θ)V sin(2θ)h, θ − ad(k−1)h = sin(2θ)V +cos(2θ)h. θ Therefore D( ) : ξ Y +ηh+ξ Y 1 1 2 2 J ad(k−1) ad(a−1)ξ Y +ηh+ad(k )ξ Y 7→ 2 t 1 1 2 2 2 (cid:26) (cid:27) = ad(k−1) ξ (cosh(t)Y sinh(t)V)+ηh+ξ Y 2 1 − 2 (cid:26) (cid:27) = ad(k−1) ξ sinh(t)V +ηh+(ξ +ξ cosh(t))Y 2 − 1 2 1 (cid:26) (cid:27) = ξ sinh(t) cos(2θ )V sin(2θ )h 1 2 2 − − (cid:18) (cid:19) +η sin(2θ )V +cos(2θ )h +(ξ +ξ cosh(t))Y 2 2 2 1 (cid:18) (cid:19) = ξ sinh(t)cos(2θ )+ηsin(2θ ) V 1 2 2 − (cid:18) (cid:19) + ξ sinh(t)sin(2θ )+ηcos(2θ ) h+ ξ +ξ cosh(t) Y. 1 2 2 2 1 (cid:18) (cid:19) (cid:18) (cid:19) To determine which (ξ ,η,ξ ) give = h + iV, we simply solve the 1 2 R linear system of equations: ξ sinh(t)cos(2θ )+ηsin(2θ ) = i 1 2 2 − ξ sinh(t)sin(2θ )+ηcos(2θ ) = 1 1 2 2 ξ +ξ cosh(t) = 0, 2 1 which has the solution: ξ = ie2iθ2csch(t) 1 − η = e2iθ2 ξ = ie2iθ2 coth(t). 2 10 JEAN BOURGAIN,ALEX KONTOROVICH,ANDPETER SARNAK Of course Y = ∂ , h = 2∂ , and Y = ∂ , whence 1 ∂θ1 ∂t 2 ∂θ2 ∂ ∂ ∂ D( ) : ie2iθ2 csch(t) +2e2iθ2 +ie2iθ2 coth(t) . J − ∂θ ∂t ∂θ 7→ R (cid:18) 1 2(cid:19) (cid:3) The formula for is derived in the same way. L 2.3. Polar Coordinates in the Disk Model. At times it will also be convenient to use polar coordinates (θ ,r,θ ) 1 2 in the unit tangent bundle of the disk model D, obtained from Cartan coordinates (θ ,t,θ ) by the change of variables 1 2 et 1 1+r r = − = tanh(t/2), = et, (2.8) et +1 1 r − with ∂ ∂r ∂ 2et ∂ 1 ∂ 1 ∂ = = = sech2(t/2) = (1 r2) , ∂t ∂t ∂r (et +1)2∂r 2 ∂r 2 − ∂r and 1 r2 1+r2 csch(t) = − , coth(t) = . 2r 2r In the (θ ,r,θ ), coordinates, the Casimir operator becomes: 1 2 1 (1 r2)2 ∂2 (1 r2)2 ∂ = − + − (2.9) 2 C 4 ∂r2 4r ∂r (1 r2)2 ∂2 ∂2 1 r4 ∂2 + − + − , 16r2 ∂θ2 ∂θ2 − 8r2 ∂θ ∂θ (cid:18) 1 2(cid:19) 1 2 and the ladder operators are 1 r2 ∂ ∂ 1+r2 ∂ = e2iθ2 i − +(1 r2) +i , (2.10) R − 2r ∂θ − ∂r 2r ∂θ (cid:18) 1 2(cid:19) and 1 r2 ∂ ∂ 1+r2 ∂ = e−2iθ2 i − +(1 r2) i . L 2r ∂θ − ∂r − 2r ∂θ (cid:18) 1 2(cid:19) 2.4. K-isotypic Vectors in the Line Model. Turning now to the line model, let = V with grading as in (2.4). s H Lemma 2.11. In the line model, if f V(2k) then 2k,s s ∈ f (x) = c(x i)k−s(x+i)−k−s (2.12) 2k,s − for some c C. ∈

Sector estimates for hyperbolic isometries

Similar Sector estimates for hyperbolic isometries