On the Borromean arithmetic orbifolds (Q6053439)

Assume \(m,n,p\) are integers \(>2\). The hyperbolic orbifold \(B_{mnp}\) is obtained from identification of the faces of a pyritohedron (a hyperbolic dodecahedron with hyperbolic right angles for all the dihedral angles but for three pairs of opposite edges which have angles \((\frac{2\pi }{m},\frac{ 2\pi }{n},\frac{2\pi }{p})\)). The identification is made by the rotations \(g_{m},g_{n},g_{p}\) on these edges. The fundamental group \( G_{mnp}\) of the hyperbolic orbifold \(B_{mnp}\) is generated by \( g_{m},g_{n},g_{p}\). In this article, the authors correct an omission in [\textit{H. M. Hilden} et al., Ohio State Univ. Math. Res. Inst. Publ. 1, 133--167 (1992; Zbl 0787.57001)] and show that \(G_{mnp}\) is arithmetic if and only if \((m,n,p)\) is one of the \(12\) triples \( (3,3,3),(3,3,\infty )\), \((3,4,4),(3,4,\infty ),(3,6,6),(3,\infty ,\infty ),(4,4,4),(4,4,\infty ),(4,\infty ,\infty )\),\((6,6,6),(6,6,\infty ),(\infty ,\infty ,\infty )\). The authors also present each arithmetic \(G_{mnp}\) as a group of \(4\times 4\) matrices with entries in the ring of integers of a totally real number field \(K\), and which are automorphs of a quaternary form \(F\) with entries in \(K\) of Sylvester type \((+,+,+,-).\)