Contemporary Mathematics
Volume 312, 2002
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 infinite
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 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.
To state our results, let
Bn
denote complex hyperbolic n-space, which can be
described concretely as either the unit ball in
en
with its Bergmann metric, or as the
set of lines in
cn+l
on which the hermitian form
h(z)
=
-lzol
2+lz112+·.
·+lznl
2 is
negative definite, with its unique (up to constant scaling factor) PU(n, I)-invariant
metric. Let A= {H1
,
H
2
,
H
3
, ... }
be a non-empty locally finite collection of totally
geodesic complex hyperplanes in
Bn.
We call
A
a complex hyperbolic arrangement
and write'}-{ for H
1
UH2UH3U· · ·. We are interested in
71"1
(Bn- 'H),
the fundamental
group of the complement. It is clear that if
A
is infinite (the case of interest here),
this group is not finitely generated. (For instance, its abelianization H
1
(Bn- '}-{, 7!.,)
is the free abelian group on the set
A.)
If
n
=
1, '}-{is a discrete subset of B
1
and
7!"1
(B1
-
'H)
is a free group, and we have nothing further to say in this case. We
thus assume throughout the paper that
n
2:
2. We say that
A
is an
orthogonal
arrangement
if any two distinct
Hi's
are either orthogonal or disjoint. The main
purpose of this note is to prove the following theorem:
Theorem
1.1.
Let A be an orthogonal complex hyperbolic arrangement. Then the
group
7!"1c
sn -
'H) has a presentation (
11, 12, ... 1 r1, r2, ... )
where each relator rk
has the form rk
=
[!i,
lij{Jlij
1],
where lij is a word in
{1, ... , lmax{i,j}- 1 .
At this point the conclusion of the theorem may seem completely technical. We
hope that a look at the proof will convince the reader that the conclusion reflects
Date: September 2, 2001.
First author partly supported by NSF grant DMS 0070930. Second and third authors partly
supported by NSF grant DMS 9900543.
©
2002 American Mathematical Society
http://dx.doi.org/10.1090/conm/312/05388
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