On discrete generalised triangle groups

Abstract

A generalised triangle group has a presentation of the formwhereRis a cyclically reduced word involving bothxandy. WhenR=xy, these classical triangle groups have representations as discrete groups of isometrics ofS2,R2,H2depending onIn this paper, for other wordsR, faithful discrete representations of these groups in Isom+H3=PSL(2,C) are considered with particular emphasis on the caseR= [x, y] and also on the relationship between the Euler characteristic χ and finite covolume representations.