Automorphic functions with respect to the fundamental group of the complement of the Borromean rings

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 1, Page 1--11.

Matsumoto, Keiji
Automorphic functions with respect to the fundamental group of the complement of the Borromean rings
We construct automorphic functions on the real $3$-dimensional hyperbolic space $\H^3$ with respect to a subgroup $B$ of $GL_2(\mathbb{Z}[i])$, which is isomorphic to the fundamental group of the complement of the Borromean rings. We utilize the pull-backs of theta functions on the hermitian symmetric domain $\D$ of type $I_{2,2}$ under an embedding from $\H^3$ into $\D$ for our construction. These automorphic functions realize the quotient space of the real $3$-dimensional upper half space by $B$ as part of an affine algebraic variety in the $6$-dimensional Euclidean space.