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ORTHOGONAL COMPLEX HYPERBOLIC ARRANGEMENTS DANIEL ALLCOCK, JAMES A. CARLSON, AND DOMINGO TOLEDO
 

Summary: ORTHOGONAL COMPLEX HYPERBOLIC ARRANGEMENTS
DANIEL ALLCOCK, JAMES A. CARLSON, AND DOMINGO TOLEDO
To Herb Clemens on his 60th birthday
1. Introduction
The purpose of this note is to study the geometry of certain remarkable in nite
arrangements of hyperplanes in complex hyperbolic space which we call orthogo-
nal arrangements : whenever two hyperplanes meet, they meet at right angles. A
natural example of such an arrangement appears in [3]; see also [2]. The concrete
theorem that we prove here is that the fundamental group of the complement of
an orthogonal arrangement has a presentation of a certain sort. As an application
of this theorem we prove that the fundamental group of the quotient of the com-
plement of an orthogonal arrangement by a lattice in PU(n; 1) is not a lattice in
any Lie group with nitely 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 nitely 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.
To state our results, let B n denote complex hyperbolic n-space, which can be
described concretely as either the unit ball in C n with its Bergmann metric, or as the
set of lines in C n+1 on which the hermitian form h(z) = jz 0 j 2 + jz 1 j 2 +    + jz n j 2 is

  

Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics