HERMITIAN POINTS IN MARKOV SPECTRA

Abstract

Let Hn 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 Bd are introduced for any d. 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. It is shown that ν2 ≤ |D|/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 ν is a Hermitian point in the Markov spectrum for SV(Z4), some discrete group of isometries of H4, then the corresponding set of extremal geodesics depends on two continuous parameters.