DIOPHANTINE APPROXIMATION IN $\mathbf{Q}(\sqrt{-5})$ AND $\mathbf{Q}(\sqrt{-6})$

Abstract

The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given. Farey polygons associated with the extended Bianchi groups Bd, d = 5, 6, are used to reduce the problem of finding the discrete part of the Markov spectrum for the group Bd to the corresponding problem for one of its maximal Fuchsian subgroup. Hermitian points in the Markov spectrum of Bd are introduced for any d. Let H3 be the upper half-space model of the three-dimensional hyperbolic space. If ν is a hermitian point in the spectrum, then there is a set of extremal geodesics in H3 with diameter 1/ν, which depends on one continuous parameter. This phenomenon does not take place in the hyperbolic plane.