HERMITIAN POINTS IN MARKOV SPECTRA

Let H(n) be the upper half-space model of the n-dimensional hyperbolic space. For n = 3, Hermitian points in the Markov spectrum of the extended Bianchi group B(d) are introduced for any d. If nu is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H(3) with diameter 1/nu, which depends on one continuous parameter. It is shown that nu(2) <= vertical bar D vertical bar/24 for any imaginary quadratic field with discriminant D, whose ideal-class group contains no cyclic subgroup of order 4, and in many other cases. Similarly, in the case of n = 4, if nu is a Hermitian point in the Markov spectrum for SV (Z(4)), some discrete group of isometries of H(4), then the corresponding set of extremal geodesics depends on two continuous parameters.