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

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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
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Abstract:
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.

Keywords: Borromean rings, hyperbolic structures, automorphic functions, theta functions.

Mathematics Subject Classification (2000): 11F55, 14P05, 57M25.

Mathematical Reviews Number: MR2223679

Received: 2005-08-04

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