Integral geometric orbifolds.

A group of matrices with entries in a number field K 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, G (p / q , (m , n)) , of the orbifolds (p / q , (m , n) are integral as subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , for all the rational knots and links p / q and all the isotropies with m > 2 , n > 2. We obtain the same result for the fundamental group G (m , m , m) of the orbifold B (m , m , m) , m > 2 , where B are the Borromean rings. The only groups G (m , n , p) with m , n , p ∈ 3 , 4 , 5 , 6 , ∞ , which are integral subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , are for the following (m , n , p) : { (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 , ∞) , (∞ , ∞ , ∞) , (3 , 3 , 5) , (3 , 5 , 5) , (3 , 5 , ∞) , (4 , 4 , 5) , (4 , 5 , ∞) , (5 , 5 , 5) , (5 , 5 , ∞) , (5 , 6 , 6) , (5 , ∞ , ∞) }. [ABSTRACT FROM AUTHOR]