Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

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\title{Crooked Planes}
\author{Todd A.~Drumm}
\author{William M.~Goldman}
\communicated_by{Gregory Margulis}
\address{
Department of Mathematics, University of Pennsylvania,  
Philadelphia, PA 19104 {\it (Drumm)}, 
Department of Mathematics, 
University of Maryland, College Park, MD  20742 {\it (Goldman)}
}
\email{tad@@math.upenn.edu {\it (Drumm),}
        wmg@@math.umd.edu {\it (Goldman)}}
\subjclass{51} 
\keywords{Space-times, affine, fundamental polyhedra, crooked planes}
\thanks{Goldman was partially supported by NSF grant DMS-8902619 and the
University of Maryland Institute for Advanced Computer Studies.
Drumm was partially supported by an NSF Postdoctoral
fellowship, and thanks Swarthmore College's KIVA project for their 
hospitality.}
\date{March 11, 1995} 
%\maketitle

\renewcommand{\o}{\operatorname}
\newcommand{\half}{{\frac12}}
\newcommand{\ppm}{I_L}
\newcommand{\SO}{{\bold{SO}}(2,1)}
\newcommand{\md}{{\bold{mid}}}
\newcommand{\Oto}{{\bold{O}}(2,1)}
\newcommand{\PPP}[1]{{\frak{P}_{#1}}}
\newcommand{\ppp}[1]{{\frak{p}_{#1}^+}}
\newcommand{\npp}[1]{{\frak{p}_{#1}^-}}
\newcommand{\SOo}{{\bold{SO}^0}(2,1)}
\newcommand{\so}{{\frak{so}}(2,1)}
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\newcommand{\vy}{{\ssfms y}}
\newcommand{\vz}{{\ssfms z}}
\newcommand{\vu}{{\ssfms u}}
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\newcommand{\rto}{{\R}^{2,1}}
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\newcommand{\xp}[1]{{\x}^+(#1)}
\newcommand{\xm}[1]{{\x}^-(#1)}
\newcommand{\xpm}[1]{{\x}^{\pm}(#1)}
\newcommand{\xmp}[1]{{\x}^{\mp}(#1)}
\newcommand{\Stem}{{\cal{S}}}
\newcommand{\Wing}{{\bold{Wing}}}
\newcommand{\nWing}{{\bold{nWing}}}
\newcommand{\Wingp}{{\cal W}^+}
\newcommand{\Wingm}{{\cal W}^-}
\newcommand{\nWingp}{{\cal M}^+}
\newcommand{\nWingm}{{\cal M}^-}
\newcommand{\CP}{{\cal C}}
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\begin{document} 

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\begin{abstract}

Crooked planes are polyhedra  
used to construct fundamental polyhedra for discrete groups of Lorentz
isometries acting properly on Minkowski (2+1)-space. These fundamental
polyhedra are regions bounded by disjoint crooked planes. We develop 
criteria for the intersection of crooked planes and apply these criteria
to proper discontinuity of discrete isometry groups.

\end{abstract}


\maketitle

%\vfill
\vskip1cm

\section{Introduction}

Let $\Gamma$ be a group of isometries of Minkowski (2+1)-space $\E$
acting freely and properly discontinuously on $\E$.  The quotient
space $\E/\Gamma$ is complete flat Lorentz space-time.  In 1983,
Margulis constructed nonamenable subgroups of the Lorentz isometry
group which act properly discontinuously on $\E$
(\cite{Margulis1},\cite{Margulis2}), realizing a suggestion of
Milnor~\cite{Milnor}.  That is, there exist complete flat Lorentz
3-manifolds $M$ whose fundamental groups are nonamenable (that is, not
virtually solvable) and these manifolds will be called {\em Margulis
space-times}.  By \cite{Drumm5} the fundamental group of a Margulis
space-time is a free group.

In~\cite{Drumm1}, complete flat Lorentz manifolds with nonabelian free
fundamental group were directly constructed using Poincar\'e's
fundamental polyhedron technique (\cite{Drumm1},\newline \cite{Drumm2}).  These
examples are diffeomorphic to solid handlebodies.  Their fundamental
groups are ``affine Schottky groups'' with fundamental domains derived
from fundamental domains of Schottky subgroups of $\SOo$ acting on the
hyperbolic plane $\Ht$.  These fundamental polyhedra are bounded by
certain disjoint polyhedral hypersurfaces in $\R^3$, called {\em crooked
planes.}

Using crooked planes, it was shown in \cite{Drumm2} that the underlying 
linear group of a free group of Lorentz isometries acting freely and properly 
discontinuously on Minkowski (2+1)-space can be any free discrete subgroup of
$\SO$.  
\begin{conj}
Every Margulis space-time has a fundamental polyhedron whose faces are
crooked planes.
\end{conj}
The classification of intersections of crooked planes is the beginning of our
program to analyze the moduli space of Margulis space-times.

%\vfill

\newpage

\section{Minkowski $2+1$-space}

Define $\rto$\ to be the 3-dimensional real vector space with the
indefinite inner product
\begin{center}
$\B(\vu,\vv )= \vu^T\cdot \ppm \cdot\vv$,
\end{center}
where $\vu$ and $\vv$
are written in terms of the usual basis $\{ \e_{1},\e_{2},\e_{3} \}$ and $\ppm$
is the diagonal matrix with entries $+1,+1,-1$.
Define the {\em Lorentzian cross-product\/} as
\begin{center}
$\vu\boxtimes\vv = \ppm \cdot\left( \vu \times \vv \right)$,
\end{center}
where $\times$ is the usual cross product. Then
$\B(\vx,\vx\boxtimes\vy) = 0$.
Let $\Oto$ be the full group of linear automorphisms of $\rto$, with 
unimodular subgroup $\SO$, and identity component $\SOo$.

For a vector $\vv$ denote its Euclidean norm by $\vert \vv \vert$, the
line containing $\vv$ by $\R\vv$, and the ray containing $\vv$ by
$\R_+\vv$.  A nonzero vector $\vv\in\rto$ is called
\begin{itemize}\samepage
\item {\em spacelike\/} if ${\B} (\vv ,\vv ) > 0$ (and 
      {\em unit-spacelike\/} if $\B(\vv,\vv) = 1$); 
\item {\em null\/} (or {\em lightlike\/}) if ${\B} (\vv ,\vv ) = 0$; 
\item {\em timelike\/}  if ${\B} (\vv ,\vv ) < 0$; 
\end{itemize} 
The {\em light cone\/} $\W$ is the set of all {\em null\/} vectors and its 
upper nappe ${\W}^{+}$ is called the {\em positive time-orientation\/}.
Similarly $\U$ is the set of {\em timelike\/} vectors  with its upper 
connected component ${\U}^{+}$ defining the positive time orientation.

For any nonzero vector $\vv\in\rto$, 
its {\em Lorentz-perpendicular plane\/} is 
\begin{equation*}
{\P}(\vv )=\{ \vu\in\rto \mid {\B} (\vu ,\vv) = 0 \} 
\end{equation*}
If $\vv$ is a spacelike vector, 
${\P}(\vv )\cap{\W}$ is the union of 2 null lines.
There exists a unique pair
\begin{equation*}
\xm{\vv} ,\xp{\vv} \in \P(\vv )\cap{\W^+} 
\end{equation*}
such that both have Euclidean norm 1 and the ordered triple 
$(\vv,\xm{\vv},\xp{\vv})$ is a positively-oriented basis of $\rto$. 

For a lightlike $\vw\in\W^+$ with Euclidean norm 1, the plane
${\P}(\vw )$ is the plane tangent to the light cone containing the
line $\R\vw$.  The null line $\R\vw$ divides ${\P}(\vw )$ into two
half-planes with closures denoted $\pp(\vw)$ and $\np(\vw)$.
Choose $\pp(\vw)$ so that $\vv\in\pp(\vw)$ when $\vw=\xp{\vv}$.
Choose $\np(\vw)$ so that $\vv\in\np(\vw)$ when $\vw=\xm{\vv}$.

Let $\E$ denote {\em Minkowski (2+1)-space,\/} the affine
space corresponding to $\rto$.  Min\-kow\-ski (2+1)-space is a simply connected
complete flat Lorentzian manifold of dimension 2+1.  For $p,q\in\E$ 
the vector $\vv = q-p$
corresponds to the translation carrying $p$ to $q$.  We also write $q
= p + \vv$.    Identify the vector space of translations of $\E$ with the
original inner product space $\rto$. Denote the group of isometries of 
$\E$ by
\begin{equation*}
{\bold K}=\Oto\ltimes\rto , 
\end{equation*}
and the homomorphism assigning to $h\in{\bold K}$ its linear part by:
\begin{equation*}
\L :{\bold H}\rightarrow\Oto .
\end{equation*}
Let $\bold{H}$ and $\bold{H}^0$ be the subgroups consisting of all 
$h\in{\bold K}$ whose linear parts are $\L(h)\in\SO$ and $\L(h)\in\SOo$,
respectively.  For isometries $h\in{\bold H}$ of $\E$, write
$h(\vx)=g(\vx)+\vv$ where $g=\L(h)\in\SO$.  The isometries $g$ and $h$
are called
\begin{itemize}\samepage
\item {\em hyperbolic\/} if $g$ has 3 distinct real eigenvalues; 
\item {\em parabolic\/} if $g$ has 1 as its only eigenvalue; 
\item {\em elliptic\/}  if $g$ has imaginary eigenvalues;
\end{itemize}
If $\vv$ is spacelike or timelike and $l= p + \R\vv$ is a line 
parallel to $\vv$, then 
$\iota_l$ denotes the inversion in line $l$.
If $\vv$ is spacelike, then $\iota_l\notin\bold H$. 

\section{Crooked planes}
Our strategy is to derive fundamental polyhedra for 
actions on Minkowski space from 
fundamental polygons for the corresponding actions on $\Ht$.
A convenient model for $\Ht$
is the hyperboloid
defined	by $\B(\vv,\vv)	= -1$, which has an induced Riemannian metric of
constant curvature -1.  Let 
$\Ht$ be the component of the hyperboloid lying in
$\U^+$.  Points of $\Ht$ correspond
to timelike lines and geodesics in $\Ht$ to spacelike planes, that is to
planes that are Lorentz perpendicular to spacelike vectors.
Suppose	that $\vv\in\rto$ is spacelike.	 Then the plane
${\P}(\vv)\subset\rto$ meets $\U$ in timelike lines determining a
geodesic in $\Ht$.

Two spacelike planes always intersect in a line.  In particular, two
ultraparallel geodesics in $\Ht$ do not intersect but the spacelike planes
corresponding to the geodesics intersect in a spacelike line. Any translations 
of these spacelike planes also intersect. We now introduce crooked planes as
objects in $\E$ that correspond to geodesics in $\Ht$.

Let $p\in\E$ be a point and $\vv\in\rto$ a spacelike vector. 
Define the {\em positively oriented crooked plane\/} $\CP(\vv,p)\subset\E$ 
with {\em vertex\/}
$p$ and {\em direction vector\/} $\vv$ to be the union of two {\em wings\/}
\begin{equation*}
\Wingp(\vv,p) = p + \pp(\xp{\vv}), \qquad
\Wingm(\vv,p) = p + \pp(\xm{\vv}) 
\end{equation*}
and the {\em stem} 
\begin{equation*} 
\Stem(\vv,p) = p +\  \{\vx\in\rto \mid\B(\vv,\vx) = 0,
\ \B(\vx,\vx) \le 0\} .
\end{equation*}


\begin{figure}[t]
\centerline{\epsfxsize=3in \epsfbox{cp1.eps}}
\caption{A crooked plane}\label{fig:cp1}
\end{figure}
\typeout{<<>>}

Define the {\em negatively oriented crooked plane} 
$\nCP(\vv,p)\subset\E$ with {\em vertex\/}
$p$ and {\em direction vector\/} $\vv$ to be the union of two {\em wings}
\begin{equation*}
\nWingp(\vv,p) = p + \np(\xp{\vv}), \qquad
\nWingm(\vv,p) = p + \np(\xm{\vv}) 
\end{equation*}
and the stem $\Stem(\vv,p)$ defined above.

For the crooked plane $\CP(\vv,p)$ (or $\nCP(\vv,p)$) there exists a
unique spacelike line $p+\R \vv$ lying on it which will be called the
{\em spine} of the crooked plane.  Given a spacelike line $\ell$ and a
point $p\in\ell$, there is a unique positively (or negatively)
oriented crooked plane with spine $\ell$ and vertex $p$.

We also define the {\em positively oriented crooked half-space\/}
$\H(\vv,p)$ and the {\em negatively oriented crooked half-space\/}
$\nH(\vv,p)$ to be the components of $\E-\C(\vv,p)$ and
$\E-\nCP(\vv,p)$, respectively, which contain
\begin{equation*}
p + \{ \vu\in\W^+ \mid \B(\vv,\vu)>0\}.
\end{equation*}
 
\section{Intersections of crooked planes}
Since two nonparallel half planes of differing orientation meet in a ray,
two oppositely oriented crooked planes always intersect:
\begin{thm}[\cite{DrummGoldman2}]\label{thm:CPnCP}
Let $\vv_1$ and $\vv_2$ be
linearly independent spacelike vectors and let $p_1,p_2\in\E$.
Then $\CP(\vv_1,p_1)\cap\nCP(\vv_2,p_2)$ is nonempty.
\end{thm}

To examine the intersection of two crooked planes of the same orientation 
we introduce the following terminology: Vectors $\vv_1,\vv_2$ are said to 
be {\em ultraparallel}, {\em asymptotic}, or {\em crossing} if
$\P(\vv_1)$ and $\P(\vv_2)$ correspond to geodesics in $\Ht$ which are 
ultraparallel, asymptotic, or crossing, respectively (compare 
\cite{DrummGoldman2}).

For crossing spacelike vectors $\vv_1$ and $\vv_2$, the 
intersection of the stems $\Stem(\vv_1,p_1)$ and $\Stem(\vv_2,p_2)$ 
contains two rays.  Since $\CP(\vv_2,p_2)$ 
and $\nCP(\vv_2,p_2)$ share a stem,
\begin{thm}[\cite{DrummGoldman2}]\label{thm:CrossingCrookedPlanes}
Let $\vv_1$ and $\vv_2$ be crossing spacelike vectors.
Then the intersection of any two crooked planes with spines 
parallel to $\vv_1$ and $\vv_2$ is nonempty.
\end{thm}


Now consider crooked planes of the same orientation whose spines are either 
ultraparallel or asymptotic. The following terminology will be useful 
since $\CP(\vv,p)=\CP(-\vv,p)$ (and $\nCP(\vv,p)=\nCP(-\vv,p)$).
Spacelike vectors $ \vv_1 , \vv_2 , \ldots , \vv_n $ are said to be
{\em consistently oriented} if $\B(\vv_i ,\vv_j ) \leq 0$ and 
$\B(\vv_i ,\xpm{\vv_j}) \leq 0$ for $i\neq j$. 
This means that the half-planes $H(\vv_i),H(\vv_j)\in\Ht$ corresponding to
$\H(\vv_i,p)$ and $\H(\vv_j,p)$ do not intersect. 

\begin{figure}[t]
\centerline{\epsfxsize=3in \epsfbox{disjoint.eps}}
\caption{Disjoint ultraparallel crooked planes}\label{fig:disjoint}
\end{figure}
\typeout{<<>>}


\begin{thm}[\cite{DrummGoldman2}]\label{thm:UltraCPCP}
Let $\vv_1$ and $\vv_2$ be consistently oriented, ultraparallel, 
unit-space\-like vectors and $p_1,p_2\in\E$. The positively oriented crooked 
planes $\CP(\vv_1,p_1)$ and $\CP(\vv_2,p_2)$ are disjoint if
and only if
\begin{equation}\label{eq:DisUltraCPCP}
\B(p_2-p_1,\vv_1\boxtimes\vv_2) >   
\vert\B(p_2-p_1,\vv_2)\vert  +
\vert\B(p_2-p_1,\vv_1)\vert .
\end{equation}
The negatively oriented crooked planes $\nCP(\vv_1,p_1)$ and 
$\nCP(\vv_2,p_2)$ are disjoint if and only if 
\begin{equation}\label{eq:DisUltranCPnCP}
\B(p_2-p_1,\vv_1\boxtimes\vv_2) <   
-(\vert\B(p_2-p_1,\vv_2)\vert  +
\vert\B(p_2-p_1,\vv_1)\vert).
\end{equation}
\end{thm}
These conditions follow from explicit description of the intersections
of the components of crooked planes. We rephrase them in terms of the
displacement vector $\vw = p_2 - p_1$ between their vertices.  Fix
spacelike vectors $\vv_1$ and $\vv_2$ and a point $p\in\E$. Consider
crooked planes with vertex $p_1 = p$ and $p_2 = p + \vw$.
For any $\vw\in\rto$, the stems $\Stem(\vv_1,p)$ and 
and $\Stem(\vv_2,p + \vw)$ are disjoint if and only if
\begin{equation*}
\vert\B(p_2-p_1,\vv_1\boxtimes\vv_2)\vert >   
\vert\B(p_2-p_1,\vv_2)\vert  +
\vert\B(p_2-p_1,\vv_1)\vert .
\end{equation*}
The set 
\begin{equation*}
A =  \left\{
\vw\in\rto \mid 
\Stem(\vv_1,p)\cap\Stem(\vv_2,p + \vw) = \emptyset
\right\}
\end{equation*}
is a disjoint union of the two open pyramids
\begin{equation*}
A_+ =  \left\{
\vw\in\rto \mid 
\CP(\vv_1,p)\cap\CP(\vv_2,p + \vw) = \emptyset
\right\}
\end{equation*}
and 
\begin{equation*}
A_- =  \left\{
\vw\in\rto \mid 
\nCP(\vv_1,p)\cap\nCP(\vv_2,p + \vw) = \emptyset
\right\}
\end{equation*}


\begin{thm}[\cite{DrummGoldman2}]\label{thm:AsymptoticCrookedPlanes}
Let $\vv_1$ and $\vv_2$ be consistently oriented, asymptotic, unit-spacelike 
vectors.  Let $\C_i=\CP(\vv_i,p_i)$ and 
$\K_i=\nCP(\vv_i,p_i)$ for $i=1,2$. The positively oriented crooked planes
$\C_1$ and $\C_2$ are disjoint if and only if
\[
\!\left\{ \begin{array}{l}
   \xm{\vv_1}=\xp{\vv_2},\\
   \B(p_2-p_1,\vv_1)<0,\\
   \B(p_2-p_1,\vv_2)<0,\\ 
   \B(p_2-p_1,\xp{\vv_1}\boxtimes\xm{\vv_2})>0;
\end{array} \! \right\}
\!\mbox{ or }  
\!\left\{ \begin{array}{l}
    \xp{\vv_1}=\xm{\vv_2},\\
    \B(p_2-p_1,\vv_1)>0,\\
	\B(p_2-p_1,\vv_2)>0,\\
	\B(p_2-p_1,\xm{\vv_1}\boxtimes\xp{\vv_2})>0.
\end{array} \! \right\}
\]
The negatively oriented crooked planes $\K_1$ and $\K_2$ are  
disjoint if and only if  
\[
\!\left\{ \begin{array}{l}
    \xm{\vv_1}=\xp{\vv_2},\\
    \B(p_2-p_1,\vv_1)>0,\\
	\B(p_2-p_1,\vv_2)>0,\\
	\B(p_2-p_1,\xp{\vv_1}\boxtimes\xm{\vv_2})<0;
\end{array} \! \right\}
\!\mbox{ or }  
\!\left\{ \begin{array}{l}
    \xp{\vv_1}=\xm{\vv_2},\\
    \B(p_2-p_1,\vv_1)<0,\\
	\B(p_2-p_1,\vv_2)<0,\\
	\B(p_2-p_1,\xm{\vv_1}\boxtimes\xp{\vv_2})<0 .
\end{array} \! \right\}
\]
\end{thm}

A pair of positively oriented ultraparallel crooked planes which 
are disjoint  are shown in Figure~\ref{fig:disjoint}.  
A pair of negatively oriented ultraparallel crooked planes with the same 
stems and translation vectors as in Figure~\ref{fig:disjoint} are 
shown in Figure~\ref{fig:intersecting} and have a nonempty 
intersection. These two figures illustrate the following more general fact.
\begin{cor}[\cite{DrummGoldman2}]\label{cor:notboth}
Let $\vv_1$ and $\vv_2$ be spacelike vectors. Either
$\CP(\vv_1,p_1)\cap\CP(\vv_2,p_2)$ or
$\nCP(\vv_1,p_1)\cap\nCP(\vv_2,p_2)$ is nonempty.
\end{cor}

\begin{figure}[t]
\centerline{\epsfxsize=3in \epsfbox{intersecting.eps}}
\caption{Intersecting ultraparallel crooked planes}\label{fig:intersecting}
\end{figure}
\typeout{<<>>}


\section{Applications}
\subsection{Groups generated by inversions}
For any spacelike line $l$ and $p\in l$,  the inversion $\iota_l$ 
preserves $\CP(\vv,p)$  leaving each wing
invariant, permuting the two components of $\Wingp(\vv,p)$ and
$\Wingm(\vv,p)$ and the two components of $\Stem(\vv,p)-\{p\}$.
Further, $\iota_l(\H(\vv,p))=\H(-\vv,p)$  so that $E-\H(\pm\vv,p)$
are fundamental domains for the
action of the group generated by the inversion in $l$.

Alternatively, $\iota_l$  also preserves $\nCP(\vv,p)$ and
$E-\nH(\pm\vv,p)$ are fundamental domains for the action of the group
generated by the inversion in $l$.

Identify $\rto$ with $\E$ via a choice of origin $o\in\E$. 
For $l= p+\R\vv$, let $\ell=\R \vv\subset\rto$ be the line parallel to $l$
through the origin. The inversion $\iota_{\ell}$  of 
$\rto$ through $\ell$
equals the reflection of $\Ht$ through the geodesic 
\begin{equation*}
S(l)={\P}(\vv)\cap \Ht = 
\CP(\vv,o)\cap\Ht = \nCP(\vv,o)\cap \Ht.  
\end{equation*}
Let $\Delta\subset\Ht$ be a convex $n$-gon whose pairs of adjacent
sides $S_1,\dots,S_n$ are ultraparallel.  $\iota_{\ell_i}$ is the
reflection in $S_i$ and let $\Gamma\subset\SO$ be the group generated
by $\iota_{\ell_1},\dots,\iota_{\ell_n}$. Then $\Gamma$ acts properly
and discretely on $\Ht$ with fundamental domain $\Delta$.  Choose 
$\vv_1,\dots,\vv_n$ to be consistently oriented so that 
\begin{equation*}
\Delta = \Ht - \left( \cup_{i=1}^n H(\vv_i) \right)
\end{equation*}

\newcommand{\gL}{\Gamma_{\bold{l}}}

An {\em affine deformation \/} of $\Gamma$ is a collection $\bold{l} =
(l_1,\dots,l_n)$ of oriented spacelike lines such that each $l_i$ has
direction vector $\vv_i$. Such an affine deformation determines an
affine representation of $\Gamma$ whose image $\gL$ is generated by
inversions $\iota_j$ in $l_j$. Lines $l\subset\E$ with direction
vector $\vv$ are parametrized by a point $p\in l$ which is determined
up to translation by $\R\vv$. Affine deformations of $\Gamma$
comprise the product
\begin{equation*}
\E/\R \vv_1 \times\dots\times  \E/\R \vv_n 
\end{equation*}
of the quotient affine spaces $\E/\R\vv_j$.

A fundamental domain for the action of $\gL$ is built from disjoint
crooked planes $\C_i = \C(\vv_i,q_i)$ with spine $l_i$ which bound a
common region.  Since a crooked plane with given spine $l$ is
determined by its vertex, the family $(\C_1,\dots,\C_n)$ is determined
by the $n$-tuple 
$\bold{q} = (q_1,\dots, q_n)\in l_1\times\dots\times l_n$.
\begin{thm}[\cite{DrummGoldman2}]\label{thm:inversion}
Let $\Gamma\subset\SO$ be as above with consistently oriented $\vv_i$'s 
parallel to the $\ell_i$'s. 
An affine 
deformation $\gL$ of $\Gamma$ is properly discontinuous if
there exist $\bold{q}\in l_1\times\dots\times l_n$ such that either 
(but not both)
of the following are true.
\begin{itemize}
\item $\C(q_1,\vv_1),\dots,\C(q_n,\vv_n)$  are disjoint and bound a 
common region 
\begin{equation*}
\E-\left(\cup_{i=1}^n \H(q_i,\vv_i)\right), 
\end{equation*}
which is a fundamental domain for the action.
\item $\nCP(q_1,\vv_1),\dots,\nCP(q_n,\vv_n)$  are disjoint and bound a 
common region 
\begin{equation*}
\E-\left(\cup_{i=1}^n \nH(q_i,\vv_i)\right), 
\end{equation*}
which is a fundamental domain for the action.
\end{itemize}
\end{thm}
These results follow directly from Theorem~3.5 of \cite{Drumm2}.
Conversely:
\begin{conj} 
Let $\Gamma\subset\SO$ be as above. An affine 
deformation $\gL$ of $\Gamma$ is properly discontinuous if and only if
there exist $\bold{q}\in l_1\times\dots\times l_n$ such that either 
(but not both) of the following are true.
\begin{enumerate}
\item $\C(q_1,\vv_1),\dots,\C(q_n,\vv_n)$  are disjoint and bound a 
common region.
\item $\nCP(q_1,\vv_1),\dots,\nCP(q_n,\vv_n)$  are disjoint and bound 
a common region.
\end{enumerate}
\end{conj}


\subsection{Groups generated by hyperbolic transformations}
All $g\in\SO$ have 1 as an eigenvalue.
For hyperbolic $g$, we define $\vx^0(g)$ to be the unique fixed 
unit-spacelike vector such that $\xm{\vx^0(g)}$ is an eigenvector with a 
corresponding eigenvalue $<1$, and  $\xp{\vx^0(g)}$ is an eigenvector with a 
corresponding eigenvalue $>1$. 

For hyperbolic $h(\vx)=g(\vx)+\vv$,  Margulis 
(\cite{Margulis1},\cite{Margulis2}) defined an invariant 
$\al(h) = \B(\vv,\vx^0(g))$.
$h$ admits a fixed point if and only if $\al(h)=0$. Further, Margulis has 
shown (\cite{Margulis1},\cite{Margulis2}, see also \cite{Drumm5}) that 
\begin{thm}[Margulis]\label{thm:oppsigns}
If $h_1$ and $h_2$ are hyperbolic Lorentz transformations such that 
$\al(h_1)$ and $\al(h_2)$ have opposite signs then any group containing 
$h_1$ and $h_2$ does not act properly on $\rto$.
\end{thm}
If there exist crooked planes $\CP(\vv_1,p_1)$ and 
$\CP(\vv_2,p_2)$  such that $h(\CP(\vv_1,p_1))= 
\CP(\vv_2,p_2)$ and $\vv_1$ and  $\vx^0(g)$ cross (equivalently, 
$\vv_2$  and  $\vx^0(g)$ cross) then the region bounded by 
$\CP(\vv_1,p_1)$ and $\CP(\vv_2,p_2)$ is a fundamental domain for the 
action of $\langle h \rangle$ on $\rto$.
The orientation of the crooked planes is related to $\al(h)$ 
by the following corollary of Theorems~\ref{thm:UltraCPCP} and 
\ref{thm:AsymptoticCrookedPlanes}:
\begin{cor} \label{cor:orientalpha}
Let $h\in{\bold H}$ be hyperbolic. If $\al(h)<0$ ($\al(h)>0$) then 
there is no fundamental domain of the action of the group generated by $h$ on 
$\E$ bounded by disjoint positively (negatively) oriented crooked 
planes.
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end