Applying the Klein model D 2 of the hyperbolic plain and identifying the geodesics in D 2 with their poles in the projective plane, the author has developed a method for determining infinite binary trees in the Markov spectrum for a Fuchsian group. In the present paper this method is applied to the groups generated by reflections in the sides of a rectangular triangle in the hyperbolic plane. The complete description of the discrete part of the Markov spectrum for any Hecke group is given.
) be an indefinite quadratic form with real coefficients and with discriminant
] is called the Markov spectrum. In 1879, Markov [9] showed, by means of continued fractions, that the set M & (1ร3, ) is discrete and it consists of the numbers (9&4รm &2 ) &1ร2 where m runs through the set of all positive integers such that (m, m 1 , m 2 ) is a solution of the Diophantine equation m 2 +m 2 1 +m 2 2 =3mm 1 m 2 . This result is closely related with the subject of Diophantine approximations (see e.g. [2] or [5]). Another development, started by Frobenius (1913) and Remak (1924), led to a new proof of Markov's theorem using the properties of quadratic forms (see
, to study the Markov spectrum, we can associate with real numerical multiples of an indefinite quadratic form f x as above the geodesic # in the hyperbolic plane H 2 =[z # C: Im z>0] whose endpoints % and %$ lie in R. This more recent point of view led to many generalizations and extensions of the original Markov theorem. The most spectacular achievement of this development is related to the parametrization of the set of simple closed geodesics on some coverings of the article no. NT972181 11 0022-314Xร97 25.00