Abstract for Orthogonal Complex Hyperbolic Arrangements

The purpose of this note is to study the geometry of certain remarkable infinite arrangements of hyperplanes in complex hyperbolic space which we call orthogonal arrangements: whenever two hyperplanes meet, they meet at right angles. A natural example of such an arrangement appears in [1]; see also [2]. The concrete theorem that we prove here is a presentation for the fundamental group of the complement of an orthogonal arrangement. As an application of this theorem we prove that the fundamental group of the complement of a certain totally geodesic divisor in the quotient of the ball by the projective unitary group with entries in the Eisenstein integers is not a lattice in any Lie group with finitely many connected components. One special case of this result is that the fundamental group of the moduli space of smooth cubic surfaces is not a lattice in any Lie group with finitely many components. This last result was the motivation for the present note, but we think that the geometry of orthogonal arrangements is of independent interest.

[1] D. Allcock. J. Carlson, and D. Toledo, The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces, Preprint 55 pp., July 8, 2000. To appear in Journal of Algebraric Geometry. (math.AG/0007048).

[2] D. Allcock. J. Carlson, and D. Toledo, Complex Hyperbolic Structure for Moduli of Cubic Surfaces, C. R. Acad. Sci. Paris, t. 326, ser I, pp 49--54, 1998 (alg-geom/970916)

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