% This is a plain tex file:
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% branched.tex
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% Daniel Allcock
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% dja2.tex
% Daniel Allcock (allcock@math.harvard.edu)
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% A set of macros for plain tex
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% LISTS
%
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% #1: the code for the citation in the file
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%--------------------------end-of-dja2.tex--------------------------
% MISC MATH STUFF
\def\({\left(}
\def\){\right)}
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\def\R{{\bbb R}} % real numbers
\def\C{{\bbb C}} % complex numbers
\let\sset=\subseteq
\let\tensor=\otimes
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%
% an \llap that you can use inside formulas without
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% computer-maintained bibliographic references
\defcite{allcock:ch4-cubic-moduli}{1}
\defcite{allcock:period-lattice-for-enriques-surfaces}{2}
\defcite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}{3}
\defcite{charney-davis:branched-covers-of-riemannian-manifolds}{4}
\defcite{charney-davis:K-pi-1-problem-for-hyperplane-complements}{5}
\defcite{ghys:sur-les-groupes-hyperboliques}{6}
\defcite{gromov:hyperbolic-groups}{7}
\defcite{gromov-thurston:pinching-constants-for-hyperbolic-manifolds}{8}
\defcite{horikawa:periods-of-enriques-surfaces-I}{9}
\defcite{horikawa:periods-of-enriques-surfaces-II}{10}
\defcite{namikawa:periods-of-enriques-surfaces}{11}
% end of computer-maintained bibliographic references
% computer-maintained cross-reference definitions
\defref{#1}{}{}{0.1}
\defref{intro}{Section}{section}{1}
\defref{sec-br-covers}{Section}{section}{2}
\defref{main-theorem}{Theorem}{theorem}{2.1}
\defref{thm-iterated-br-cover}{Theorem}{theorem}{2.2}
\defref{punctured-balls-are-path-connected}{Lemma}{lemma}{2.3}
\defref{sec-apps}{Section}{section}{3}
\defref{thm-hyperplane-complements-aspherical}{Theorem}{theorem}{3.1}
\defref{thm-neighborhoods-described}{Lemma}{lemma}{3.2}
\defref{thm-weak-homotopy-equiv}{Lemma}{lemma}{3.3}
\defref{cubic-surfaces}{Corollary}{corollary}{3.4}
\defref{enriques-surfaces}{Corollary}{corollary}{3.5}
% end of computer-maintained cross-reference definitions
% stuff for intro
\def\cp{\C P} % complex projective space
\def\ch{\C H} % complex hyperbolic space
\def\moduli{{\cal M}} % moduli space
\def\PGL{{\rm PGL}} % projective linear group
\def\U{{\rm U}} % unitary group
% stuff for background
\def\x{\kappa} % quick greek letter
\def\M{M_{\x}^2} % comparison space
\def\sx{\sqrt\x} % abbrev
% stuff for simple branched covers
\def\Xhat{{\widehat X}} % base space
\def\Yhat{{\widehat Y}} % branched cover
\def\D{\Delta} % branch locus
\def\Dtil{\tilde{\D}} % preimage of branch locus
\def\cat{CAT($\x$)} % abbreviation
\def\diam{\mathop{\rm Diam}\nolimits} % diameter of a set
\def\G{\Gamma} % a homotopy
\def\btil{\tilde{\b}} % a lifted path
\def\dtil{\tilde{\d}} % a lifted path
\def\ytil{\tilde y} % a lifted point
\def\geo#1{\overline{#1}} % geodesic joining 2 points
\def\tri#1{\triangle#1} % geodesic triangle
\def\Util{\tilde{U}} % a ball in the cover
\def\cA{({\bf A})} % 7 cases in a proof
\def\cB{({\bf B})}
\def\cC{({\bf C})}
\def\cD{({\bf D})}
\def\cE{({\bf E})}
\def\cF{({\bf F})}
\def\cG{({\bf G})}
% stuff for the proof that triangles with edge in \D satisfy \cat
\def\Tbar{\bar{T}}
\def\Ubar{\bar{U}}
\def\Kbar{\bar{K}}
\def\Bbar{\bar{B}}
\def\abar{\bar{a}}
\def\Abar{\bar{A}}
\def\atil{\tilde{\a}}
\def\xbar{\bar{x}}
% stuff for iterated covers
\def\S{{\cal S}} % union of vector subspaces
\def\H{{\cal H}} % union of submanifolds
\def\Htil{{\widetilde{\cal H}}} % preimage of union of submanifolds
\def\s{\sigma} % generator of \pi_1
\def\Mhat{{\hat M}} % base manifold
\def\Nhat{{\hat N}} % branched cover
\def\pihat{{\hat\pi}} % branched covering map
\def\xtil{{\tilde x}} % point in Ntil above x
\def\ytil{{\tilde y}} % point in Ntil above y
\def\ztil{{\tilde z}} % point in Ntil above z
\def\ctil{{\tilde\gamma}} % path in Ntil
\def\Vtil{{\widetilde V}} % a neighborhood of \xtil
\def\Btil{{\widetilde B}} % a closed neighborhood of \xtil
\def\Atil{{\widetilde A}} % a branched cover of a standard disk
\def\Gammatil{{\widetilde\Gamma}} % lifted homotopy
% stuff for applications
\def\w{\omega} % a cube root of 1
\def\E{{\cal E}} % eisenstein numbers
\def\I{{\cal I}} % the hyperplanes passing through a given point
\def\DD{{\cal D}} % symmetric space for O(2,10)
% misc stuff
\def\a{\alpha}
\def\b{\beta}
\def\c{\gamma}
\def\d{\delta}
\def\e{\varepsilon}
\def\image{\mathop{\rm image}\nolimits}
%-----------------Beginning-of-Document---------------------------
\begin{titleblock}
\title {Asphericity of moduli spaces via curvature}
% \title {Metric curvature of infinite branched covers}
\author {Daniel Allcock}
\support {Supported in part by an NSF Postdoctoral Fellowship}
\date {26 March 2001}
% previous versions
%
% nov 13, 2000
% october 15, 2000
% July 24, 2000
% 25 May 1999
% 14 May 1999
% 2 May 1999
% 14 Feb 1999
% 9 Oct 1998
\address {Department of Mathematics, Harvard University,
Cambridge, MA 02138}
\email {allcock@math.harvard.edu}
\homepage{http://www.math.harvard.edu/$\sim$allcock}
\subject {53C23 (14J28, 57N65)}
\keywords{branched cover, ramified cover, Alexandrov space,
cubic surface, Enriques surface}
% \note {}
\end{titleblock}
\begin{abstract}
We show that
under suitable conditions a branched cover satisfies the same upper
curvature bounds as its base space. First we do this when the
base space is a metric space satisfying
Alexandrov's curvature condition \cat\ and the branch locus is
complete and convex. Then we treat branched covers of
a Riemannian manifold over suitable mutually
orthogonal submanifolds. In neither setting do we require that the
branching be locally finite. We apply our results to hyperplane
complements in several Hermitian symmetric spaces of nonpositive
sectional curvature in order to prove that two moduli spaces arising
in algebraic geometry are aspherical. These are the moduli spaces of
the smooth cubic surfaces in $\cp^3$ and of the
smooth complex Enriques surfaces.
\end{abstract}
\begin{section}{intro}{Introduction}
It is well-known that taking branched covers
usually introduces negative curvature. One
can see this phenomenon in elementary examples using Riemann
surfaces, and the idea also plays a role in the construction
\cite{gromov-thurston:pinching-constants-for-hyperbolic-manifolds}
of exotic manifolds with negative sectional curvature. In this
paper we work in the setting of Alexandrov's comparison geometry;
for background see
\cite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}.
In this setting we will establish
the persistence of upper curvature bounds in
branched covers.
A simple way to build a cover $\Yhat$ of a space $\Xhat$
branched over $\D\sset\Xhat$ is to take any covering space $Y$
of $\Xhat-\D$ and define $\Yhat=Y\cup\D$. We call $\Yhat$ a
simple branched cover of $\Xhat$ over $\D$. Our first result,
\ref{main-theorem}, states that if $\Xhat$
satisfies Alexandrov's
\cat\ condition and $\D$ is complete and satisfies a convexity
condition then the natural
metric on $\Yhat$ also satisfies \cat.
%We include examples showing that an important
%completeness hypothesis cannot be dropped; our examples also
%disprove several claims in the literature.
The question which motivated this investigation is whether the
moduli space of smooth cubic surfaces in $\cp^3$ is aspherical
(i.e., has contractible universal cover). It is, and
our argument also establishes the analogous result for the
moduli space of smooth complex Enriques surfaces. To prove these
claims, we use the fact that each of these moduli spaces is
known to be covered by a Hermitian symmetric space with
nonpositive sectional curvature, minus an arrangement of complex
hyperplanes. In each case the hyperplanes have the property that
any two of them are orthogonal wherever they meet. In
\ref{sec-apps} we show that such a hyperplane complement
is aspherical. The result
(\ref{thm-hyperplane-complements-aspherical}) is more general
because the symmetric space structure is not needed.
Theorem~5.3 of
\cite{charney-davis:branched-covers-of-riemannian-manifolds}
morally contains this result and also suggests a substantial
generalization of it. However, there are some difficulties with the proof.
If $\Mhat$ is the symmetric space and $\H$ is the union of the
hyperplanes, then the idea is to apply standard nonpositive
curvature techniques like the Cartan-Hadamard theorem to the
universal cover $N$ of $M=\Mhat-\H$. The problem is that $N$ is
not metrically complete. One can pass to its metric completion
$\Nhat$, but this introduces problems of its own. First there is
the issue of how $N$ and $\Nhat$ are related. We resolve this by
a simple trick that shows that the inclusion $N\to\Nhat$ is a
homotopy equivalence. The second problem is that $\Nhat$ is not
a manifold and not even locally compact, so that one cannot use
the techniques of Riemannian geometry. But it is still a metric
space and it turns out to satisfy Alexandrov's CAT(0) condition
locally. It is then easy to show that $N$ and $\Nhat$
are contractible.
In order to understand this curvature bound for $\Nhat$,
the reader should imagine a closed ball $B$ in
$\C^n$, equipped with
some Riemannian metric, minus the coordinate
hyperplanes. The metric completion of the universal cover of the
hyperplane complement can be obtained by first taking a simple
branched cover of $B$ over one hyperplane, then taking a
simple branched cover of this branched cover over (the preimage
of) the second hyperplane, and so on. If the hyperplanes are
mutually orthogonal and totally geodesic then \ref{main-theorem}
may be used inductively to bound the curvature of the iterated
branched cover. Observe that the base space of each branched cover
fails to be locally compact except in the first step.
This means that the inductive argument
requires a theorem treating branched covers of
spaces more general than manifolds.
We also note that the condition of mutual orthogonality in branched
covers has appeared before, for example for the
modified Deligne complexes of
\cite{charney-davis:K-pi-1-problem-for-hyperplane-complements},
which are certain metric polyhedral complexes of piecewise
constant curvature. In fact, in this polyhedral setting our results are
already well-established. We need to go beyond the piecewise
constant curvature case for the applications to algebraic
geometry.
For the reader's convenience we recall some definitions from
\cite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}. $D_\x$ is the diameter of the simply-connected complete
surface of constant sectional curvature $\x$. A metric space $X$
is $D_\x$-geodesic if any two points at distance~$0$, together with the point $(1,0)$. Let $\D$ be
% the positive $y$-axis. Then $\Xhat$ is a convex subset of
% $\R^2$, hence CAT(0), and $\D$ is a closed convex
% subset of $\Xhat$. The set $X=\Xhat-\D$ is contractible, so any
% cover of it is a union of disjoint copies of it. Taking $Y$ to
% be the cover with 2 sheets, $\Yhat$ is isometric to the
% upper half plane in $\R^2$ together with the points
% $(\pm1,0)$. There is no geodesic joining these two points, so
% $\Yhat$ is not a geodesic space. This provides
% a counterexample to several assertions in the literature, such
% as \ecite{gromov:hyperbolic-groups}{4.3--4.4},
% \ecite{januszkiewicz:hyperbolizations}{Lemma~1.1} and
% \ecite{davis91:hyperbolization_of_polyhedra}{Lemma~2.4}.
% \end{example}
% \begin{example}{Example}
% Although the space $\Yhat$ of the previous example is not
% geodesic, it still has curvature $\leq0$. The following example shows
% that even this may fail if $\D$ is not complete. We take $\Xhat$
% to be the set of points $(x,y,z)$ of $\R^3$ whose first nonzero
% coordinate is positive, together with the origin. That is,
% $\Xhat$ is the union of an open half-space together with an open
% half-plane in its boundary, together with a ray in {\it its}
% boundary. We take $\D$ to be the set of points of $\Xhat$ with
% vanishing $x$-coordinate, which is the union of the open half-plane
% and the ray. Then $\Xhat$ is a convex
% subset of $\R^3$ and $\D$ is closed and convex in
% $\Xhat$. As before, any cover of $X=\Xhat-\D$ is a union of copies
% of $X$, and we take $Y$ to be the cover with 2 sheets. Then
% $\Yhat$ is isometric to the subset of $\R^3$ given by
% $$
% \Yhat=\D\cup\set{(x,y,z)\in\R^3}{x\neq0},
% $$
% equipped with the path metric induced by the Euclidean
% metric. It is easy to see that for each $n\geq1$ the points
% $(\pm1/n,-1/n,-1/n)$ are joined by no geodesic of
% $\Yhat$. Since every neighborhood of $0$ contains such a pair of
% points, $0$ has no geodesic neighborhood.
% \end{example}
\begin{proof}{Proof}
We have obtained a proof in full generality, but here we make
the additional assumptions that $\x\leq0$ and $\Xhat$ is
complete. This is sufficient for our applications. The idea is
to realize $\Yhat$ as a Gromov-Hausdorff limit of spaces which
are obviously \cat. For $\e>0$ let $\D_\e$ be the closed
$\e$-neighborhood of $\D$, and let $\Yhat_\e$ be obtained by
gluing together a copy of $\D_\e$ and a copy of $Y$, so that all
preimages in $Y$ of any given point of $\D_\e-\D$ are
identified. This space has a natural path metric---in fact it
is a simple branched cover of $\Xhat$ over $\D_\e$. It is
obvious that $\Yhat$ is a Gromov-Hausdorff limit of the
$\Yhat_\e$. Since $\Yhat$ is complete it suffices by
\ecite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}{Cor.~3.10}
to show that each $\Yhat_\e$ is \cat. Since $\Yhat_\e$ is complete
and simply connected, it suffices to show that $\Yhat_\e$ is locally
\cat. If $x\in\Yhat_\e$ lies at distance${}\neq\e$ from $\D$ then $x$
admits a \cat\ neighborhood because it has a neighborhood isometric to
an open subset of $\Xhat$. Now suppose $d(x,\D)=\e$. We let $U$ be
the closed $\e/2$-ball about the image of $x$ in $\Xhat$. Then $x$
admits a neighborhood which is the union of some number of copies of
$U$, glued together along $U\cap\D_\e$. This neighborhood of $x$ is
\cat\ by Reshetnyak's lemma, since $U\cap\D_\e$ is complete and is
convex in $U$.
\end{proof}
Next we define precisely what we mean by a branched
cover which is locally an iterated branched cover of a manifold
over a family of mutually orthogonal totally geodesic submanifolds. Then
we show that such a branched cover satisfies the
same upper bounds on local curvature as the base manifold. We
prove this only in the case of nonpositive curvature, and
indicate what else is needed in the general case.
We say that a finite set $\{S_1,\ldots,S_n\}$ of codimension-2 subspaces of an
even-dimensional real vector space $A$ is normal if it is
equivalent to some $n$ of the $m$ coordinate hyperplanes in
$\C^m$ under an $\R$-linear isomorphism $A\to\C^m$. If $A$ is
odd-dimensional then we call $\{S_1,\ldots,S_n\}$ normal if it is equivalent
to the products with $\R$ of some $n$ of the $m$ coordinate hyperplanes
of $\C^m$, in
$\C^m\times\R$. We write $\S$ for $\cup_iS_i$.
Now suppose $\H_0$ is a family of immersed submanifolds of a
Riemannian manifold $\Mhat$ with union $\H$. We say that $\H_0$ is normal at
$x\in\Mhat$ if there is a set $\{S_1,\ldots,S_n\}$ of mutually orthogonal
subspaces of $T_x\Mhat$ that are normal in the sense above and
have the following property. We require that there be an open
ball $U$ about $0$ in $T_x\Mhat$ which the exponential map
carries diffeomorphically onto its image $V$, such that
$V\cap\H=\exp_x(U\cap\S)$, and such that each $\exp_x(S_i\cap U)$ is a convex
subset of $V$.
We say that $\H_0$ is normal if it is normal at
each $x\in\Mhat$. In this case, each element of $\H_0$ is
totally geodesic and intersections of elements of $\H_0$ are orthogonal.
If $x\in\Mhat$ then
$\pi_1(V-\H)\isomorphism\pi_1(U-\S)\isomorphism\pi_1(T_x\Mhat-\S)\isomorphism\Z^n$.
The first two isomorphisms are obvious and the
last follows from the fact that $T_x\Mhat-\S$ is a product of $n$ punctured
planes and a Euclidean space.
We choose generators $\s_1,\ldots,\s_n$ for
$\pi_1(T_x\Mhat-\S)$ by taking a representative for $\s_i$ to be
a simple circular loop that links $S_i$ but none of the other
$S_j$. We say that a connected covering space of $T_x\Mhat-\S$
is standard if the subgroup of $\Z^n$ to which it
corresponds is generated by $\s_1^{d_1},\ldots,\s_n^{d_n}$ for some
$d_1,\ldots,d_n\in\Z$. We apply the same
terminology to the corresponding cover of $V-\H$.
In particular, the universal cover is
standard. An arbitrary covering space of $V-\H$ is called
standard if each of its components is.
We write $M$ for $\Mhat-\H$. If $\pi:N\to M$ is a covering space
then we say that $N$ is a standard cover of $M$ if for each
$x\in\Mhat$ with $V$ as above, $\pi:\pi^{-1}(V-\H)\to V-\H$ is a
standard covering in the sense above. In this case, we take
$\Nhat$ to be a certain subset of the metric completion of $N$,
namely those points which map into $\Mhat$ under the extension
of $\pi$. In particular, if $\Mhat$ is complete then $\Nhat$ is
the completion of $N$. We denote the natural extension
$\Nhat\to\Mhat$ of $\pi$ again by $\pi$, and call $\Nhat$ a
standard branched covering of $\Mhat$ over $\H_0$. The simplest
example of a standard branched cover is $\pi:\C^n\to\C^n$,
carrying $(z_1,\ldots,z_n)$ to
$(z_1^{d_1},\ldots,z_n^{d_n})$. We have just extended this by
making the definition local and allowing infinite branching.
\begin{theorem}{thm-iterated-br-cover}
If a Riemannian manifold $\Mhat$
has sectional curvature bounded above by $\x\leq0$ and
$\pi:\Nhat\to\Mhat$ is a standard branched cover over a
normal family $\H_0$ of immersed submanifolds of $\Mhat$, then
$\Nhat$ is locally \cat.
\end{theorem}
\begin{lemma}{punctured-balls-are-path-connected}
Let $X$ be a length space with metric completion $\Xhat$. Then
every open ball in $\Xhat$ meets $X$ in a path-connected set.
\end{lemma}
\begin{proof}{Proof}
Suppose given an open ball $U$ about $x\in\Xhat$, and $y,z\in U\cap
X$. Choose $x'\in X$ near $x$ and join $y$ and $z$ to $x'$ by paths
in $X$ that are short enough that they are forced to lie in
$U$.
\end{proof}
\begin{proof}{Proof of~\ref{thm-iterated-br-cover}}
We will write $\Htil$ for $\pi^{-1}(\H)$. Suppose
$\xtil\in\Nhat$ lies over $x\in\Mhat$ and let $S_1,\ldots,S_n$,
$U$ and $V$ be
as in the definition of the normality of $\H_0$ at $x$.
Let $r$ be the common radius of
$U$ and $V$.
We write $T_i$
for $\exp_x(U\cap S_i)\sset V$. It is clear that geodesics from
$\xtil$ to nearby points are lifts of radial geodesics
from $x$. By choosing $r$ small
enough we may suppose that $\pi^{-1}(V)$ is the disjoint union
of the $r$-balls about the points of $\pi^{-1}(x)$. We also
choose $r$ small enough so that $V$ and all smaller balls
centered at $x$ are convex. We write $\Vtil$ for the open
$r$-ball about $\xtil$; \ref{punctured-balls-are-path-connected} assures us that $\Vtil-\Htil$ is
a connected covering space of $V-\H$. Taking generators
$\sigma_1,\ldots,\sigma_n$ for $\pi_1(V-\H)$ as above, the
standardness of the cover assures us that the covering
$\Vtil-\Htil\to V-\H$ corresponds to the subgroup generated by
$\sigma_1^{d_1},\ldots,\sigma_n^{d_n}$ for some
$d_1,\ldots,d_n$. We take $B$ (resp. $\Btil$) to be the closed
$r'$-ball about $x$ (resp. $\xtil$), where we will choose $r'0$. The projection maps $B\to B\cap T_j$ may
increase distances in the presence of positive curvature. All
that is important for us is that the length of a path in $B$ {\it
with endpoints in $T_k$} does not increase under projection to
$T_k$. Even this is not true, but we only need the result for
paths of length $<2r'$. One should choose $r'$ small enough so
that any path in $B$ of length $<2r'$, with endpoints in $T_k$,
grows no longer under the projection to $T_k$. Presumably this
can be done but I have not checked the details.
\end{remark}
\Ref{thm-iterated-br-cover} has been widely believed,
but this seems to be the first proof. As mentioned before, it is morally
contained in theorem~5.3 of
Charney and Davis
\cite{charney-davis:branched-covers-of-riemannian-manifolds},
who consider locally finite branched covers of Riemannian
manifolds over subsets more complicated than
mutually orthogonal submanifolds. Unfortunately there are gaps
in their proof which I do not know how to bridge.
(Lemma~5.7 does
not seem to follow from lemma~5.6. Also, the techniques of \cite{ghys:sur-les-groupes-hyperboliques}
referred to in passing to finish the proof of theorem~5.3 use
properties of Riemannian manifolds, like continuous dependence
of sufficiently short geodesics on their endpoints, that are not
established for branched covers.)
Nevertheless
their infinitesimal
\cat\ condition (condition~3 of theorem~5.3)
is very natural, and their theorem surely
holds and extends to the case of locally infinite branching.
% %% PROOF IN FULL GENERALITY
% What follows is a proof of \ref{main-theorem} in full
% generality. We sometimes write $[x,y]$ for a
% geodesic joining points $x$ and $y$.
% \begin{proof}{Proof that $\Yhat$ is $D_\x$-geodesic}
% Suppose $x,y\in\Yhat$ lie at distance~$0$, and $\H$ may be
taken to be the union of the (complex) hyperplanes in $\DD$ which are
the orthogonal complements of the norm $-1$ vectors of
$L$. We regard the moduli space of Enriques surfaces as being
the orbifold $(\DD-\H)/\Gamma$.
\begin{corollary}{enriques-surfaces}
The moduli space $(\DD-\H)/\Gamma$ has contractible orbifold
universal cover.
\end{corollary}
\begin{proof}{Proof}
The proof that hyperplanes that meet do so orthogonally is the same as
before. Local finiteness follows by essentially the same standard
argument: any $v\in\DD$ defines a 2-dimensional positive-definite
subspace $V$ of $L\tensor\R$, namely the projection to $L\tensor\R$ of
the complex line it represents. For each $N$ there are only finitely
many $r\in L$ with $r\cdot r=-1$ having projection to $V$ of
norm~$\leq N$. This proves local finiteness and hence normality. Then
we use the fact that $\DD$ has nonpositive sectional curvature and
appeal to \ref{thm-hyperplane-complements-aspherical}.
\end{proof}
\end{section}
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\bye