CONJUGACY CLASSES OF GAMMA(2) AND SPECTRAL RIGIDITY

The free group Gamma(2) is generated by A = (1 2, 0 1) and B = (1 0, -2 1),and setting chi((xi,eta))(A) = exp(2 pi i xi), chi((xi,eta))(B) = exp(2 pi i eta) defines a unitary character on Gamma(2) for 0 less than or equal to xi, eta < 1. A program is devised to compute mu(tr) = Sigma chi((xi,eta))(conj. class), summed over all primitive conjugacy classes of Gamma(2) of trace tr. Combined with a Luo-Sarnak theorem, this yields lower bounds for the spectral variance for a large sampling of characters in the 0 < xi,eta < 1. The results indicate that the Berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems. The program is also used compute theta(x) = Sigma 1n(N({gamma})), summed over all primitive conjugacy classes of Gamma(2) of norm N({gamma}) less than or equal to x. The function theta(x) is asymptotic to x, and the remainder can be written as \theta(x) - x\ = x(beta), The values of beta(x) are computed for all traces between 3202 and 4802 (here x = tr(2) -2). The p's cluster around 0.6, attaining a maximum of 2/3. Finally, it is proved that the remainder theta(x) - x has a negative bias by showing that the mean normalized remainder converges to a negative limit.