Arithmeticity of orbifold generalised triangle groups
\textit{C. Maclachlan} and \textit{G. J. Martin} [J. Reine Angew. Math. 511, 95-117 (1999; Zbl 0960.20029)], initiated a programme to investigate 2-generator arithmetic Kleinian groups. They proved that only finitely many arithmetic Kleinian groups can be generated by 2 elliptic elements, and they classified these groups in the non-cocompact case. In the paper under review, the authors investigate the cocompact case, restricting to a class of generalized triangle groups admitting a presentation \[ \langle a,b\mid a^l=b^m=w^n=1\rangle, \] where \(l,m,n\geq 2\) and where \(w\) is a cyclically reduced word in the free product on \(\{a,b\}\). These groups were considered before by Jones and Reid, and they arise as fundamental groups of 3-dimensional orbifolds. The authors address the question as to which generalized triangle groups admit faithful representations as cocompact arithmetic Kleinian groups. They give a table containing a list of \(l\), \(m\) and \(n\) together with conditions on the word \(w\) for which a generalized triangle group admits a faithful representation as an arithmetic Kleinian group. They also give a description of the corresponding orbifold.
scientific article; zbMATH DE number 1878465
scientific article