Conjugacy classes of $\Gamma (2)$ and spectral rigidity

Abstract:

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 $0 \leq \xi ,\eta < 1$. A program is devised to compute \[ \mu ({\text {tr}}) = \sum {{\chi _{(\xi ,\eta )}}({\text {conj}}. {\text {class}}),} \] 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 range $0 < \xi ,\eta < 1$. 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 to compute \[ \theta (x) = \sum {\ln (N(\{ \gamma \} )),} \] summed over all primitive conjugacy classes of $\Gamma (2)$ of norm $N(\{ \gamma \} ) \leq x$. 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 $x = {\text {tr}^2} - 2$). The $\beta$’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.