On the Borromean arithmetic orbifolds

Abstract

We revisit the fundamental groups Gmnp of the orbifolds Bmnp, where the underlying manifold is the 3-sphere S3 and the Borromean rings are the singular set with isotropies of order m, n and p. We correct an omission in [2] and show that Gmnp is arithmetic if and only if (m,n,p) is one of the 12 triples (3,3,3), (3,3,∞), (3,4,4), (3,4,∞), (3,6,6), (3,∞,∞), (4,4,4), (4,4,∞), (4,∞,∞), (6,6,6), (6,6,∞), (∞,∞,∞). The main purpose of the paper is to present each Gmnp, arithmetic, as a group of 4×4 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 (+,+,+,−).