Addendum for

**Orthogonal Complex Hyperbolic Arrangements**

with J. Carlson and D. Toledo;
*Symposium in Honor of C. H. Clemens*, *Contemp. Math.* **312** (2002) 1-8

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In the proof of lemma 3.1 we make an assertion which is true but not
sufficiently justified in the paper. The setting is that
**Gamma** is a lattice in **PU(n,1)** for **n>1**, carrying a
hyperplane arrangement to itself, **H**_{i} is one of these
hyperplanes, and **Gamma**_{i} is its stabilizer. In the
paper we use without comment the fact that **Gamma**_{i} is
a lattice in the automorphism group **PU(n-1,1)** of
**H**_{i}.

There may be a simple way to see this, but we don't know of one. The
truth of the assertion follows from very general theorems in ergodic
theory. See M. Ratner's survey,
Invariant measures and orbit closures for unipotent actions on
homogeneous spaces, *Geom. Funct. Anal.* **4** (1994),
236-257, and its references, for background. Briefly, the first few
pages of this article, up through conjecture 3 (which is proven
later), imply the following:

**Theorem:**
Suppose **X** is a symmetric space, **Gamma** is a lattice in
**Aut(X)**, and **Y** is a symmetric subspace of **X**
corresponding to a subgroup of **Aut(X)** that is generated by
unipotent elements. Then the closure of the image of **Y** in
**Gamma\X** is the quotient of a symmetric subspace **Z** of **X** modulo a
lattice in its stabilizer.

In our case, we take **X** to be complex hyperbolic **n**-space
and **Y** to be the hyperplane **H**_{i}. The
hypothesis **n>1** implies that **PU(n-1,1)** is generated by
unipotent elements. So we apply the theorem; since **Y** is a
hyperplane, **Z** is either a hyperplane or all of
**CH**^{n}. The latter case is impossible because the
arrangement is not dense in **CH**^{n}. In the former case
it is easy to see that **Z** is equivalent to **H**_{i}
under **Gamma**, and it follows from the theorem that
**Gamma**_{i} is a lattice in **Aut(H**_{i}), as
desired.

The story behind the omission of this detail is that an
earlier version of the paper treated a special class of hyperplane
arrangements, rather than the more general version given in the
published paper. For these special arrangements, the fact that
**Gamma**_{i} is a lattice in **PU(n-1,1)** was obvious.