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