Integral geometric orbifolds

Abstract

A group of matrices with entries in a number field [Formula: see text] is integral if it has a finite index subgroup of matrices whose entries are algebraic integers. In this paper, we show that the fundamental groups, [Formula: see text], of the orbifolds [Formula: see text] are integral as subgroups of both [Formula: see text] and of [Formula: see text], for all the rational knots and links [Formula: see text] and all the isotropies with [Formula: see text] [Formula: see text]. We obtain the same result for the fundamental group [Formula: see text] of the orbifold [Formula: see text], [Formula: see text], where [Formula: see text] are the Borromean rings. The only groups [Formula: see text] with [Formula: see text], which are integral subgroups of both [Formula: see text] and of [Formula: see text] are for the following [Formula: see text]: [Formula: see text]