Farey neighbours, modular symbols and divergent geodesics

We give effective asymptotic counting results for pairs of Farey neighbours and for modular symbols in $\mathbb{Q}$, in imaginary quadratic number fields and in definite quaternion algebras over $\mathbb{Q}$, using the distribution of common perpendiculars between Margulis cusp neighbourhoods and divergent geodesics in hyperbolic manifolds. We describe the tangency properties of the canonical Margulis cusp neighbourhoods in Bianchi hyperbolic $3$-orbifolds.