| Let Gamma ⊂ SL(2, Z ) be a non-elementary Fuchsian group of the second kind, i.e. a finitely-generated subgroup having infinite index in SL(2, Z ) and infinite limit set. Then the volume of the Riemann surface Gamma H is infinite, where H is the Poincare model of the hyperbolic plane. Let delta = delta(Gamma) ½. Let Gamma(N) denote the infinite-volume principal "congruence" subgroup of Gamma of level N, Gamma(N) = {gamma ∈ Gamma : gamma ≡ I(N)} and let Spec(Gamma( N) H ) denote the spectrum of the Laplace-Beltrami operator on the Hilbert space L2(Gamma(N) H ). Assume that for some theta satisfying delta > theta > ½ we have a spectral gap: SpecGN \H∩&sqbl0; 0,q1-q &parr0;=SpecG 1\H ∩&sqbl0;0,q1-q &parr0;, uniformly in square-free N..;For gamma ∈ Gamma, let f(gamma) = 1/ Jm (gamma -1 ) so that f( ab cd ) = c2 + d 2. Let Gammainfinity be the stabilizer of f, Gamma infinity = {gamma ∈ Gamma : gamma = +/-1* 0+/-1 }, which we assume is nontrivial. We show that the set of values of f is thin, in the sense that ;The spectral gap exists in general if delta > 5/6 with theta = 5/6, a fact due to Gamburd [14]. In particular, if delta > 149/150, then f contains infinitely-many 25-almost primes outside of B . |
