%_ **************************************************************************
%_ * The TeX source for AMS journal articles is the publishers TeX code     *
%_ * which may contain special commands defined for the AMS production      *
%_ * environment.  Therefore, it may not be possible to process these files *
%_ * through TeX without errors.  To display a typeset version of a journal *
%_ * article easily, we suggest that you retrieve the article in DVI,       *
%_ * PostScript, or PDF format.                                             *
%_ **************************************************************************
% Author Package file for use with AMS-LaTeX 1.2
\controldates{24-MAY-2005,24-MAY-2005,24-MAY-2005,24-MAY-2005}
 
\RequirePackage[warning,log]{snapshot}
\documentclass{era-l}
\issueinfo{11}{05}{}{2005}
\dateposted{May 27, 2005}
\pagespan{40}{46}
\PII{S 1079-6762(05)00145-9}
\usepackage{amsmath}
\usepackage{latexsym}
\usepackage{mathrsfs}
\usepackage{graphicx}

\copyrightinfo{2005}{American Mathematical Society}
\revertcopyright

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}


\newcommand{\myH}{\mathbb{H}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\CC}{\mathcal C}
\newcommand{\LL}{\mathcal L}
\newcommand{\myaa}{\alpha}
\newcommand{\myo}{\text{O}}
\newcommand{\reflec}{\text{Reflec}}
\newcommand{\po}{\text{PO}}
\newcommand{\aut}{\text{Aut}}
\newcommand{\myvol}{\text{vol}}
\newcommand{\covol}{\text{covol}}
\newcommand{\isom}{\text{Isom}\,}

\begin{document}
\title{The smallest hyperbolic 6-manifolds}
\author{Brent Everitt}
\address{Department of Mathematics, University of York, York
YO10 5DD, England}
\email{bje1@york.ac.uk}
\thanks{The first author is grateful to the Mathematics Department, Vanderbilt
University for its hospitality during a stay when the results of this paper
were obtained.}

\author{John Ratcliffe}
\address{Department of Mathematics, Vanderbilt University, Nashville, TN 37240}
\email{ratclifj@math.vanderbilt.edu}

\author{Steven Tschantz}
\address{Department of Mathematics, Vanderbilt University, Nashville, TN 37240}
\email{tschantz@math.vanderbilt.edu}
\subjclass{Primary 57M50}
\commby{Walter Neumann}
\date{October 31, 2004}
\begin{abstract}
By gluing together copies of an all right-angled Coxeter polytope 
a number of open hyperbolic $6$-manifolds
with Euler characteristic $-1$ are constructed. They are the first known
examples of hyperbolic $6$-manifolds having the smallest possible volume. 
\end{abstract}
\maketitle

\section{Introduction}

The last few decades have seen a surge of activity in the study of
finite volume hyperbolic manifolds---that is, 
complete Riemannian $n$-manifolds of constant sectional curvature $-1$. 
Not surprisingly for geometrical objects, volume has been, and continues to be, the most 
important invariant for understanding their sociology.
The possible volumes in a fixed dimension form a well-ordered subset of $\R$,
indeed a discrete subset except in $3$ dimensions (where the orientable manifolds at
least have ordinal 
type $\omega^\omega$). Thus it is a natural problem with a long history to construct
examples of manifolds with minimum volume in a given dimension.

In $2$ dimensions the solution is classical, with the minimum volume
in the compact orientable case achieved by a genus $2$ surface, and in the 
noncompact
orientable case by a once-punctured torus or thrice-punctured sphere
(the identities of the manifolds are of course also known in the nonorientable case). 
In $3$ dimensions the compact
orientable case remains an open problem with the Matveev-Fomenko-Weeks 
manifold \cite{Matveev-Fomenko88, Weeks85} obtained
via $(5,-2)$-Dehn surgery on the sister of
the figure-eight
knot complement conjecturally the smallest. 
Amongst the noncompact orientable $3$-manifolds the figure-eight knot complement realizes
the minimum volume \cite{Meyerhoff01}, and the Gieseking manifold (obtained by 
identifying the sides of a regular hyperbolic tetrahedron as in 
\cite{Everitt02b, Prok98})
does so for the nonorientable ones \cite{Adams87}.
One could also add ``arithmetic'' to our list of adjectives and so have eight
optimization problems to play with (so that  
the Matveev-Fomenko-Weeks manifold is known
to be the minimum volume orientable, arithmetic compact $3$-manifold; see 
\cite{Chinburg01}).

When $n\geq 4$ the picture is murkier, although in even dimensions we have
recourse to the Gauss-Bonnet Theorem, so that in particular the minimum volume
a $2m$-dimensional hyperbolic manifold could possibly have, is when the 
Euler characteristic $\chi$ satisfies $|\chi|=1$. 
The first examples of noncompact $4$-manifolds with $\chi=1$ were constructed
in \cite{Ratcliffe00} (see also \cite{Everitt02}). The compact case remains a 
difficult unsolved problem, although if we restrict to arithmetic manifolds,
then it is known \cite{Belolipetsky02, Conder04} that a minimum volume arithmetic compact orientable 4-manifold $M$
has $\chi \leq 16$ and $M$ is isometric to the orbit space of a torsion-free subgroup 
of the hyperbolic Coxeter group $[5,3,3,3]$. 
The smallest compact hyperbolic $4$-manifold currently known to exist 
has $\chi=8$ and is constructed in \cite{Conder04}.
Manifolds of very small volume have been constructed in $5$ dimensions 
\cite{Everitt02, Ratcliffe04},
but the smallest volume $6$-dimensional example hitherto known has $\chi=-16$ 
\cite{Everitt02}.

In this paper we announce the discovery of a number of noncompact nonorientable 
hyperbolic $6$-manifolds with Euler characteristic $\chi=-1$. The method of construction
is classical in that the manifolds are obtained by identifying the sides of a 
$6$-dimensional hyperbolic Coxeter polytope.

\section{Coxeter polytopes}

Let $C$ be a convex (not necessarily bounded) polytope of finite volume in a simply
connected space $X^n$ of constant curvature. Call $C$ a Coxeter polytope if the 
dihedral angle subtended by two intersecting $(n-1)$-dimensional sides 
is $\pi/m$ for some integer $m\geq 2$. When $X^n=S^n$ or the Euclidean space $E^n$,
such polyhedra have been completely classified \cite{Coxeter34}, 
but in the hyperbolic space 
$H^n$, a complete classification remains a difficult problem 
(see for example \cite{Vinberg85} and the references there).

If $\Gamma$ is the group generated by reflections in the $(n-1)$-dimensional sides
of $C$, then $\Gamma$ is a discrete cofinite subgroup of the Lie group $\isom X^n$,
and every discrete cofinite reflection group in $\isom X^n$ arises in this way 
from some Coxeter polytope, which is uniquely defined up to isometry. 
The Coxeter symbol for $C$ (or $\Gamma$) has nodes
indexed by the $(n-1)$-dimensional sides, and an edge labeled $m$ joining the nodes
corresponding to sides that intersect with angle $\pi/m$ (label the edge joining
the nodes of nonintersecting sides by $\infty$). In practice the labels $2$ and $3$
occur often, so that edges so labeled are respectively removed or left unlabeled.

Let $\Lambda$ be an $(n+1)$-dimensional Lorentzian lattice, 
that is, an $(n+1)$-dimensional 
free $\Z$-module equipped with a $\Z$-valued bilinear form of signature $(n,1)$. 
For each $n$, there is a unique such $\Lambda$, denoted
$I_{n,1}$, that is odd and self-dual (see \cite[Theorem V.6]{Serre73},
or \cite{Milnor73, Neumaier83}).
By \cite{Borel62}, the
group $\myo_{n,1}\Z$ of automorphisms of $I_{n,1}$ acts 
discretely, cofinitely by isometries on the hyperbolic space $H^n$ obtained by 
projectivising the negative norm vectors
in the Minkowski space-time $I_{n,1}\otimes\R$ (to get a faithful action one normally
passes to the centerless version $\po_{n,1}\Z$). 

Vinberg and Kaplinskaya showed \cite{Vinberg78, Vinberg72} that
the subgroup $\reflec_n$ of $\po_{n,1}\Z$ generated by reflections in positive norm vectors
has finite index if 
and only if $n\leq 19$, thus yielding a family of cofinite reflection groups and 
corresponding finite volume Coxeter polytopes
in the hyperbolic spaces $H^n$ for $2\leq n\leq 19$. 
Indeed, Conway and Sloane showed (\cite[Chapter 28]{Conway93} or 
\cite{Conway82})
that for $n\leq 19$ the quotient 
of $\po_{n,1}\Z$ by $\reflec_n$ is a subgroup of the automorphism group of the Leech lattice.
Borcherds \cite{Borcherds87} showed that the (non-self-dual) even sublattice of $I_{21,1}$
also acts cofinitely, yielding the highest-dimensional example known of a Coxeter group
acting cofinitely on hyperbolic space.

When $4\leq n\leq 9$ the group $\Gamma=\reflec_n$ has Coxeter symbol,
$$\includegraphics{era145el-fig-1}$$
with $n+1$ nodes and $C$ a noncompact, finite volume
$n$-simplex $\Delta^n$ (when $n>9$, the polytope $C$ has a more complicated 
structure).

Let $v$ be the vertex of $\Delta^n$ opposite the side $F_1$ marked on the symbol,
and let $\Gamma_v$ be the stabilizer in $\Gamma$ of this vertex. This stabilizer
is also a reflection group with symbol as shown, and is finite for 
$4\leq n\leq 8$ (being the Weyl group of type $A_4,D_5,E_6,E_7$ and $E_8$ respectively)
and infinite for $n=9$ (when it is the affine Weyl group of type $\widetilde{E}_8$). Let
$$
P_n=\bigcup_{\gamma\in\Gamma_v} \gamma(\Delta^n),
$$
a convex polytope obtained by gluing $|\Gamma_v|$ copies of the simplex $\Delta^n$ together. 
Thus, $P_n$ has finite volume precisely when $4\leq n\leq 8$, although it is noncompact, 
with a mixture of finite vertices in $H^n$ and cusped ones on $\partial H^n$.
In any case, $P_n$ is an all right-angled Coxeter
polytope: its sides meet with dihedral angle $\pi/2$ or are disjoint.
This follows immediately from the observation that the sides of $P_n$ arise from the
$\Gamma_v$-images of the side of $\Delta^n$ opposite $v$, and this side intersects the 
other sides of $\Delta^n$ in dihedral angles $\pi/2$ or $\pi/4$. 
Vinberg has conjectured that $n=8$
is the highest dimension in which finite volume all right-angled polytopes exist in 
hyperbolic space.

The volume of the polytope $P_n$ is given by 
$$\myvol(P_n)=|\Gamma_v|\myvol(\Delta^n)=|\Gamma_v|[\po_{n,1}\Z:\Gamma]\covol(\po_{n,1}\Z),$$
where $\covol(\po_{n,1}\Z)$ is the volume of a fundamental region for the action
of $\po_{n,1}\Z$ on $H^n$ (and for $4\leq n\leq 9$ the index $[\po_{n,1}\Z:\Gamma]=1$). 
When $n$ is even, we have by \cite{Siegel36} and 
\cite{Ratcliffe97},
$$
\covol(\po_{n,1}\Z)=\frac{(2^{\frac{n}{2}}\pm 1)\pi^{\frac{n}{2}}}{n!}
\prod_{k=1}^{\frac{n}{2}} |B_{2k}|,
$$
with $B_{2k}$ the $2k$-th Bernoulli number and with the plus sign if 
$n\equiv 0,2\mod 8$ and the minus sign otherwise. 

Alternatively (when $n$ is even), we have
recourse to the Gauss-Bonnet Theorem, so that $\myvol(P_n)=\kappa_n|\Gamma_v|
\chi(\Gamma)$, 
where $\chi(\Gamma)$ is the Euler characteristic of
the Coxeter group $\Gamma$ and $\kappa_n=2^n (n!)^{-1} (-\pi)^{n/2} (n/2)!$.
The Euler characteristic of Coxeter groups can be easily computed from their symbol
(see \cite{Chiswell92, Chiswell76} or \cite[Theorem 9]{Everitt02}). Indeed, when
$n=6$, 
$\chi(\Gamma)=-1/\LL$ where $\LL=2^{10}\,3^4\,5$ and so 
$\myvol(P_6)=8\pi^3|E_6|/15\LL=\pi^3/15$.

The Coxeter symbol for $P_n$ has a nice description in terms of finite reflection groups.
If $v'$ is the vertex of $\Delta^n$ opposite the side $F_2$, let $\Gamma_e$ be the 
pointwise stabilizer of $\{v,v'\}$: the elements thus stabilize the edge $e$ of 
$\Delta^n$ joining $v$ and $v'$. 

Now consider the Cayley graph $\CC_v$ for 
$\Gamma_v$ with respect to the generating reflections in the sides of the symbol
for $\Gamma_v$. Thus, $\CC_v$ has vertices in one-to-one correspondence with the elements
of $\Gamma_v$ and for each generating reflection $s_{\myaa}$, an undirected edge 
labeled $s_{\myaa}$ connecting vertices $\gamma_1$ and
$\gamma_2$ if and only if $\gamma_2=\gamma_1s_{\myaa}$ in $\Gamma_v$. In particular,
$\CC_v$ has $s_2$ labeled edges corresponding to the reflection in $F_2$. Removing these
$s_2$-edges
decomposes $\CC_v$ into components, each of which is a copy of the Cayley graph $\CC_e$ for
$\Gamma_e$, with respect to the generating reflections. 

Take as the nodes of the symbol for $P_n$ these connected components. If two components
have an $s_2$-labeled edge running between any two of their vertices in $\CC_v$, then leave 
the corresponding nodes
unconnected; otherwise, connect them by an edge labeled $\infty$. The resulting symbol
(respectively the polytope $P_n$) thus has $|\Gamma_v|/|\Gamma_e|$ nodes 
(resp.~sides). 
The number of sides of $P_n$ for $n=4,5,6,7,8$ is $10,16,27,56$ and $240$ respectively. 

\section{Constructing the manifolds}

We now restrict our attention to the case $n=6$. 
We work in the hyperboloid model of hyperbolic 6-space
$$H^6=\{x\in \R^7: x_1^2+x_2^2+\cdots+x_6^2-x_7^2=-1\ \hbox{and}\ x_7>0\}$$
and represent the isometries of $H^6$ by Lorentzian $7\times 7$ matrices that 
preserve $H^6$. 
The right-angled polytope $P_6$ has 27 sides each congruent to $P_5$.  
We position $P_6$ in $H^6$ so that 6 of its sides are bounded 
by the 6 coordinate hyperplanes $x_i=0$ for $i=1,\ldots, 6$  
and these 6 sides intersect at the center $e_7$ of $H^6$. 
Let $K_6$ be the group of 64 diagonal Lorentzian $7\times 7$ matrices 
${\rm diag}(\pm 1,\ldots,\pm 1,1)$. 
The set $Q_6=K_6P_6$, which is the union of 64 copies of $P_6$, 
is a right-angled convex polytope with 252 sides. 
We construct hyperbolic 6-manifolds, with $\chi =-8$,
by gluing together the sides of $Q_6$ by a proper side-pairing 
with side-pairing maps of the form $rk$ with $k$ in $K_6$ and $r$ 
a reflection in a side $S$ of $Q_6$. 
The side-pairing map $rk$ pairs the side $S'=kS$ to $S$ 
(see \S 11.1 and \S 11.2 of \cite{Ratcliffe94} 
for a discussion of proper side-pairings). 
We call such a side-pairing of $Q_6$ simple. 
We searched for simple side-pairings of $Q_6$ that yield 
a hyperbolic 6-manifold $M$ with a freely acting $\Z/8$ symmetry group 
that permutes the 64 copies of $P_6$ making up $M$ in such a way 
that the resulting quotient manifold is obtained by gluing 
together 8 copies of $P_6$. 
Such a quotient manifold has $\chi = -8/8=-1$. 
This is easier said than done, since the search space 
of all possible side-pairings of $Q_6$ is very large. 
We succeeded in finding desired side-pairings of $Q_6$ by employing a strategy 
that greatly reduces the search space. 
The strategy is to extend a side-pairing in dimension 5 with the desired 
properties to a side-pairing in dimension 6 with the desired properties. 

Let $Q_5 = \{x\in Q_6: x_1 = 0\}$. 
Then $Q_5$ is a right-angled convex 5-dimensional polytope with 72 sides. 
Note that $Q_5$ is the union $K_5P_5$ of 32 copies of $P_5$ where 
$P_5= \{x\in P_6: x_1=0\}$ and  
$K_5$ is the group of 32 diagonal Lorentzian $7\times 7$ matrices 
${\rm diag}(1,\pm 1,\ldots,\pm 1,1)$.  
A simple side-pairing of $Q_6$ that yields a hyperbolic 6-manifold $M$ 
restricts to a simple side-pairing of $Q_5$ 
that yields a hyperbolic 5-manifold which is a totally geodesic hypersurface of $M$. 
All the orientable hyperbolic 5-manifolds that are obtained by gluing together 
the sides of $Q_5$ by a simple side-pairing are classified in 
\cite{Ratcliffe04}. 

We started with the hyperbolic 5-manifold $N$, numbered 27 in 
\cite{Ratcliffe04},  
obtained by gluing together the sides of $Q_5$ by the simple side-pairing 
with side-pairing code {\tt 2B7JB47JG81}. 
The manifold $N$ has a freely acting $\Z/8$ symmetry group 
that permutes the 32 copies of $P_5$ making up $N$ in such a way that 
the resulting quotient manifold is obtained by gluing together 4 copies of $P_5$. 
A generator of the $\Z/8$ symmetry group of $N$ is represented by the Lorentzian $6\times 6$ matrix
$$
\left(\begin{array}{cccccc}
     \phantom{-}1 & 0 &\phantom{-}0 & \phantom{-}1 & 0 &     -1 \\
     \phantom{-}0 & 0 &\phantom{-}0 & \phantom{-}0 & 1 & \phantom{-}0 \\
         -1 & 0 &          -1 & \phantom{-}0 & 0 & \phantom{-}1 \\
     \phantom{-}0 & 1 &\phantom{-}0 & \phantom{-}0 & 0 & \phantom{-}0 \\
     \phantom{-}0 & 0 &          -1 &     -1 & 0 & \phantom{-}1 \\
         -1 & 0 &          -1 &     -1 & 0 & \phantom{-}2
        \end{array} \right).
$$
The strategy is to search for simple side-pairings of $Q_6$ that yield 
a hyperbolic 6-manifold with a freely acting $\Z/8$ 
symmetry group with generator represented by the following Lorentzian $7\times 7$ matrix 
that extends the above Lorentzian $6\times 6$ matrix: 
$$
\left(\begin{array}{ccccccc}
     \phantom{-}1 &\phantom{-}0 & 0 &\phantom{-}0 & \phantom{-}0 & 0 &\phantom{-}0 \\
     \phantom{-}0 &\phantom{-}1 & 0 &\phantom{-}0 & \phantom{-}1 & 0 &     -1 \\
     \phantom{-}0 &\phantom{-}0 & 0 &\phantom{-}0 & \phantom{-}0 & 1 & \phantom{-}0 \\
          \phantom{-}0 &-1 & 0 &          -1 & \phantom{-}0 & 0 & \phantom{-}1 \\
     \phantom{-}0 &\phantom{-}0 & 1 &\phantom{-}0 & \phantom{-}0 & 0 & \phantom{-}0 \\
     \phantom{-}0 &\phantom{-}0 & 0 &          -1 &     -1 & 0 & \phantom{-}1 \\
         \phantom{-}0 &-1 & 0 &          -1 &     -1 & 0 & \phantom{-}2
        \end{array} \right).
$$      
For such a side-pairing the resulting quotient manifold can be obtained 
by gluing together 8 copies of $P_6$ by a proper side-pairing. 
By a computer search we found 14 proper side-pairings of 8 copies of
$P_6$ in this way, and hence we found 14 hyperbolic $6$-manifolds with $\chi = -1$. 
Each of these 14 manifolds is noncompact with 
volume $8\myvol(P_6)=8\pi^3/15$ and five cusps. 
These 14 hyperbolic $6$-manifolds represent at least 7 different isometry types, 
since they represent 7 different homology types. 
Table 1 lists side-pairing codes for 7 simple side-pairings of $Q_6$ 
whose $\Z/8$ quotient manifold has homology groups isomorphic to  
$\Z^a\oplus(\Z/2)^b\oplus(\Z/4)^c\oplus(\Z/8)^d$ for nonnegative integers $a,b,c,d$
encoded by $abcd$ in the table. In particular, all 7 manifolds in Table 1 
have a finite first homology group. 


All of our examples, with $\chi = -1$, can be realized as the orbit space $H^6/\Gamma$ 
of a torsion-free subgroup $\Gamma$ of $\po_{6,1}\Z$ of minimum index. 
These manifolds are the first examples of hyperbolic 6-manifolds 
having the smallest possible volume. 
All these manifolds are nonorientable. 
In the near future, we hope to construct orientable examples
of noncompact hyperbolic 6-manifolds having $\chi= -1$.
\begin{table}[t]
\caption{Side-pairing codes and homology groups of the seven examples.}
\begin{center}
\begin{tabular}{lllllll}
$N$&$SP$&\ \ $H_1$&\ \ $H_2$&\ \ $H_3$&\ \ $H_4$&\ \ $H_5$\\
&&$\phantom{\mathbb Z}$0248&$\phantom{\mathbb Z}$0248&$\phantom{\mathbb Z}$0248&
$\phantom{\mathbb Z}$0248&$\phantom{\mathbb Z}$0248\\
1&{\tt GW8dNEEdN4ZJO1k2l1PIY}&
\phantom{${\mathbb Z}$}0401&
\phantom{${\mathbb Z}$}1910&
\phantom{${\mathbb Z}$}4821&
\phantom{${\mathbb Z}$}1500&
\phantom{${\mathbb Z}$}0000\\
2&{\tt HX9dNFEcM5aKU6f3f6UKa}&
\phantom{${\mathbb Z}$}0401&
\phantom{${\mathbb Z}$}1810&
\phantom{${\mathbb Z}$}8710&
\phantom{${\mathbb Z}$}5500&
\phantom{${\mathbb Z}$}0000\\
3&{\tt HX9dNFEcM5YIO1l3l1OIY}&
\phantom{${\mathbb Z}$}0401&
\phantom{${\mathbb Z}$}2900&
\phantom{${\mathbb Z}$}7810&
\phantom{${\mathbb Z}$}4400&
\phantom{${\mathbb Z}$}1000\\
4&{\tt HX9dNFEcM5YIO6l3l6OIY}&
\phantom{${\mathbb Z}$}0401&
\phantom{${\mathbb Z}$}2800&
\phantom{${\mathbb Z}$}7910&
\phantom{${\mathbb Z}$}4400&
\phantom{${\mathbb Z}$}1000\\
5&{\tt HX9dNFEcM5YIOxl3lyOIY}&
\phantom{${\mathbb Z}$}0211&
\phantom{${\mathbb Z}$}2800&
\phantom{${\mathbb Z}$}4821&
\phantom{${\mathbb Z}$}1400&
\phantom{${\mathbb Z}$}1000\\
6&{\tt HX9dNFEcM5YIOyl3lxOIY}&
\phantom{${\mathbb Z}$}0211&
\phantom{${\mathbb Z}$}2800&
\phantom{${\mathbb Z}$}4930&
\phantom{${\mathbb Z}$}1400&
\phantom{${\mathbb Z}$}1000\\
7&{\tt HX9dNFEcM5aKUxf3fyUKa}&
\phantom{${\mathbb Z}$}0301&
\phantom{${\mathbb Z}$}1900&
\phantom{${\mathbb Z}$}5630&
\phantom{${\mathbb Z}$}2500&
\phantom{${\mathbb Z}$}0000\\
\end{tabular}
\end{center}
\end{table}

\begin{thebibliography}{10}

\bibitem{Adams87} C.~Adams, {\em The noncompact hyperbolic $3$-manifold of
minimum volume}, Proc. AMS {\bf 100} (1987), 601--606. \MR{0894423 (88m:57018)}

\bibitem{Belolipetsky02} M.~Belolipetsky, {\em On volumes of arithmetic
quotients of $SO(1,n)$}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear).

\bibitem{Borcherds87} R.~Borcherds, {\em Automorphism groups of Lorentzian
lattices}, J. Algebra {\bf 111} (1987), no. 1, 133--153. \MR{0913200 (89b:20018)}

\bibitem{Borel62} A.~Borel and Harish-Chandra, {\em Arithmetic subgroups of
algebraic groups}, Ann. of Math. {\bf 75} (1962), no. 3, 485--535. \MR{0147566
(26:5081)}

\bibitem{Chinburg01} T.~Chinburg, E.~Friedman, K.~N.~Jones, and A.~W.~Reid,
{\em The arithmetic hyperbolic 3-manifold of smallest volume}, Ann. Scuola
Norm. Sup. Pisa Cl. Sci. {\bf 30} (2001), no. 4, 1--40. \MR{1882023
(2003a:57027)}

\bibitem{Chiswell92}  I.~M.~Chiswell, {\em The Euler characteristic of graph
products and Coxeter groups}, in Discrete Groups and Geometry, W.~J.~Harvey and
Colin Maclachlan (Editors), London Math. Soc. Lect. Notes, vol. 173, 1992, 
pp. 36--46.
\MR{1196914 (94a:05090)}

\bibitem{Chiswell76}  I.~M.~Chiswell, {\em Euler characteristics of groups},
Math. Z. {\bf 147} (1976), 1--11. \MR{0396785 (53:645)}

\bibitem{Conder04}  M.~D.~E.~Conder and C.~Maclachlan, {\em Small volume
compact hyperbolic $4$-manifolds}, to appear, Proc. Amer. Math. Soc.

\bibitem{Conway93} J.~Conway and N.~J.~A.~Sloane, {\em Sphere Packings,
Lattices and Groups}, Second Edition, Springer, 1993. \MR{1194619 (93h:11069)}

\bibitem{Conway82}  J.~Conway and N.~J.~A.~Sloane, {\em Leech roots and Vinberg
groups}, Proc. Roy. Soc. London Ser. A {\bf 384} (1982), no. 1787, 233--258.
\MR{0684311 (84b:10048)}

\bibitem{Coxeter34}  H.~S.~M.~Coxeter, {\em Discrete groups generated by
reflections}, Ann. of Math. {\bf 35} (1934), no. 2, 588--621. \MR{1503182}

\bibitem{Davis85}  M.~W.~Davis, {\em A hyperbolic 4-manifold}, Proc. Amer.
Math. Soc. {\bf 93} (1985), 325--328. \MR{0770546 (86h:57016)}

\bibitem{Everitt02} B.~Everitt, {\em Coxeter groups and hyperbolic manifolds},
Math. Ann. {\bf 330} (2004), no. 1, 127--150. \MR{2091682}

\bibitem{Everitt02b} B.~Everitt, {\em $3$-Manifolds from Platonic solids}, Top.
App. {\bf 138} (2004), 253--263. 2035484 (2004m:57031)

\bibitem{Ratcliffe05} B.~Everitt, J.~Ratcliffe and S.~Tschantz, {\em Arithmetic
hyperbolic $6$-manifolds of smallest volume}, (in preparation).

\bibitem{Matveev-Fomenko88} V.~Matveev and A.~Fomenko, {\em Constant energy
surfaces of Hamilton systems, enumeration of three-dimensional manifolds in
increasing order of complexity, and computation of volumes of closed hyperbolic
manifolds}, Russian Math. Surveys {\bf 43} (1988), 3--24. \MR{0937017
(90a:58052)}

\bibitem{Meyerhoff01} C.~Cao and G.~R.~Meyerhoff, {\em The orientable cusped
hyperbolic $3$-manifolds of minimum volume}, Invent. Math. {\bf 146} (2001),
no. 3, 451--478. \MR{1869847 (2002i:57016)}

\bibitem{Milnor73} J.~Milnor and D.~Husemoller, {\em Symmetric Bilinear Forms},
Springer, 1973. \MR{0506372 (58:22129)}

\bibitem{Neumaier83} A.~Neumaier and J.~J.~Seidel, {\em Discrete Hyperbolic
Geometry}, Combinatorica {\bf 3} (1983), 219--237. \MR{0726460 (85h:51025)}

\bibitem{Prok98} I.~Prok, {\em Classification of dodecahedral space forms},
Beitr\"{a}ge Algebra Geom. {\bf 39} (1998), no. 2, 497--515. \MR{1642716
(99i:52029)}

\bibitem{Ratcliffe94} J.~Ratcliffe, {\em Foundations of hyperbolic manifolds},
Graduate Texts in Mathematics 149, Springer 1994. \MR{1299730 (95j:57011)}

\bibitem{Ratcliffe97}  J.~Ratcliffe and S.~Tschantz, {\em Volumes of integral
congruence hyperbolic manifolds}, J. Reine Angew. Math. {\bf 488} (1997),
55--78. \MR{1465367 (99b:11076)}

\bibitem{Ratcliffe00}  J.~Ratcliffe and S.~Tschantz, {\em The volume spectrum
of hyperbolic 4-manifolds}, Experiment. Math. {\bf 9} (2000), no. 1, 101--125.
\MR{1758804 (2001b:57048)}

\bibitem{Ratcliffe04}  J.~Ratcliffe and S.~Tschantz, {\em Integral congruence
two hyperbolic $5$-manifolds}, Geom. Dedicata {\bf 107} (2004), 187--209.
\MR{2110762}

\bibitem{Serre73} J.-P.~Serre, {\em A Course in Arithmetic}, Graduate Texts in
Mathematics {\bf 7}, Springer, 1973. \MR{0344216 (49:8956)}

\bibitem{Siegel36}  C.~L.~Siegel, {\em \"{U}ber die analytische Theorie der
quadratischen Formen II}, Ann. Math. {\bf 37} (1936), 230--263. \MR{1503276}

\bibitem{Vinberg85} E.~B.~Vinberg, {\em Hyperbolic reflection groups}, Uspekhi
Mat. Nauk {\bf 40} (1985), no. 1, 29--66; English transl., Russian Math. 
Surveys {\bf 40}
(1985), no. 1, 31--75. \MR{0783604 (86m:53059)}

\bibitem{Vinberg78} E.~B.~Vinberg and I.~M.~Kaplinskaya, {\em On the groups
$\textup{O}_{18,1}(\mathbb Z)$ and $\textup{O}_{19,1} (\mathbb Z)$}, Dokl.
Akad. Nauk. SSSR {\bf 238} (1978), no. 6, 1273--1275; English transl., Soviet
Math. Dokl. {\bf 19} (1978), no. 1, 194--197. \MR{0476640 (57:16199)}

\bibitem{Vinberg72} E.~B.~Vinberg, {\em On the groups of units of certain
quadratic forms}, Math. Sb. {\bf 87} (1972), no. 1, 18--36; English transl.,
Math. USSR, Sb. {\bf 16} (1972), no. 1, 17--35. \MR{0295193 (45:4261)}

\bibitem{Weeks85} J.~Weeks, {\em Hyperbolic structures on $3$-manifolds}, Ph.D.
thesis, Princeton University, 1985. 

\end{thebibliography}

\end{document}