Problems

Abstract

In the last paragraph we have constructed elements of Г (for a special case only, but it is obvious how to generalize the procedures), but no attempt was made to prove that the elements so obtained actually generate Г or at least a subgroup of finite index. To the best of my knowledge, this has been achieved for no such Г, if d ≥ 3. Paradoxically enough, generation of a cofinite subgroup becomes trivial if we pass from Λ to M_n (Λ), n ≥ 2: by results of K-theory, the elementary matrices will do the job; see e.g. [BRi] and the references quoted there. In the quaternion case d = 2, an algorithm is given in Fricke-Klein [FK], as well as many examples ([FK], p. 539 ss); marvellously drawn are the tesselations of the upper half plane resulting from the explicit fundamental domains and their translates.