Distribution of modular symbols in $\mathbb{H}^3$

Abstract: We introduce a new technique for the study of the distribution of modular symbols, which we apply to congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ its ring of integers, then for certain congruence subgroups of $\mathrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups, where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and the second moment of the distribution.