AMBIGUOUS NUMBERS UNDER THE ACTION OF THE GROUP

Abstract

We discuss here some basic properties of certain real quadratic irrational numbers a called ambiguous numbers, under the action of G= <X==/>by using coset diagrams. We show that if u is totally positive then y(a), y (a), y' (a) are totally negative. Also the linear fractional transformation x transforms totally positive number to a totally negative number and vice-vera and an ambiguous number into an ambiguous number. We show also that if uis an ambiguous number then one of y(a) v (a),y' (a) is ambiguous and the other two are totally negative. Finally. we show that a GCQ(n), where Q (Vn)=(a+wa.c?Zhn... =1 and the C action of G on the projective line over Q (Vn) is always intransitive.