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\begin{document}
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\begin{center}
\bf $c$ - extensions of $P$- and $T$-geometries
\end{center}
\medskip
\begin{center}{\sc
Gernot Stroth, Corinna Wiedorn\\
Martin-Luther-Universit\"at Halle-Wittenberg\\
Institut f\"ur Algebra und Geometrie\\
06099 Halle, Germany}
\end{center}
\bigskip
We consider affine extensions of geometries of Petersen and of tilde type,
i.e., of flag-transitive geometries $\Gamma $ belonging to
a diagram
$$\hbox{\Diagn{1}{2}{n-2}{n-1}{n}{$X$},}$$
where either
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$=$
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i.e., the geometry of edges and vertices of the Petersen graph,
or
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$=$
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i.e., the 3-fold cover of the generalized $Sp_4(2)$-quadrangle.
We call such geometries
$c$-$P$- resp. $c$-$T$-geometries because the residue of an element
of type 1 is a $P$- resp. $T$-geometry. Such geometries have been
classified by different authors and an overview over the results is
given in \cite{IS1}. In particular, there exist only two
$P$-geometries of rank 3 with automorphism groups $Aut(M_{22})$
resp. $3Aut(M_{22})$ and three $T$-geometries of rank 3 with automorphism
groups
$M_{24}$, $He$, and $3^7Sp_6(2)$. In this paper we will assume:
\begin{enumerate}
\item[($*$)] If $F$ is a flag of $\Gamma $ of type $\{ 1,\ldots ,n-3\}$ then
the residue $res(F)$ of $F$ in $\Gamma $ is the $P$-geometry for
$M_{22}$ or the $T$-geometry for $M_{24}$.
\end{enumerate}
Since there are no $P$-geometries of rank $\ge 5$ and no
$T$-geometries of rank $\ge 5$ satisfying ($*$), this forces
$n\le 6$. On the other hand, it is known that for $n=3$ the universal
cover of $\Gamma $ is infinite, and
the case that $\Gamma $ is a $c$-$P$-geometry of rank 4 satisfying ($*$)
has been considered in \cite{AC}. There it is shown that
$Aut(\Gamma )$ is isomorphic to a factorgroup of one of
the groups $2^{11}\!\!n :\! Aut(M_{22})$, $M_{24}$ or $2\!\cdot\!
U_6(2)\! :\! 2$ (where by $H\! :\! K$ and $H\!\cdot\! K$ we denote as
usually a split- resp. non-split extension of a group $K$ by a group $H$).
In
the first case, the normal 2-subgroup is the universal representation group
(see below) of the $P$-geometry for $M_{22}$. It is isomorphic to a
submodule of index two (the even half) of the Golay co-code.
Models for the other two geometries are described in \cite{AC}.
\newtheorem{defi}{Definition}
\begin{defi}[see \cite{IPS}] Let $\Gamma $ be any geometry which
contains some sets of
objects ${\cal P}$ and ${\cal L}$ (``points'' and ``lines'') such that
any line $l\in {\cal L}$ is incident to exactly three points of ${\cal P}$.
Let $U(\Gamma )$ be the abstract group defined by
$$
\begin{array}{rl}
U(\Gamma )=&\langle u_x|x\in {\cal P}, u_x^2=1=u_xu_yu_z \hbox{ whenever
$x,y,z$ are the }\\
&\hbox{ three points incident to a line } l\in{\cal L}\rangle .
\end{array}
$$
Then $U(\Gamma )$ is called the {\bf universal representation group} of the
point-line system $({\cal P},{\cal L})$.\par
If $\Gamma $ is a $P$- or $T$-geometry with diagram
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\put(1,2.5){\scriptsize {1}}
\put(11,2.5){\scriptsize{2}}
\put(21,2.5){\scriptsize{n-2}}
\put(31,2.5){\scriptsize{n-1}}
\put(41,2.5){\scriptsize{n}}
\end{picture}
we define the group $U(\Gamma )$ to be the universal representation group of
the point-line system $({\cal P},{\cal L})$ where $\cal P$ and $\cal L$ are the objects of $\Gamma $ of type 1 and 2.
\end{defi}
Of course, the group $U(\Gamma )$ might become trivial, as it does
e.g. in case of the
rank 4 $P$-geometry for $M_{23}$. But if it is nontrivial, then
$Aut(\Gamma )$ acts on $U(\Gamma )$ in a natural way and the
semidirect product of $U(\Gamma )$ with $Aut(\Gamma )$ acts
flag-transitively on a $c$-extension of $\Gamma $.\par
The universal representation groups of the $P$- and $T$-geometries
occurring as residues in this paper are all known (see \cite{IPS}). For the
$P$-geometry for $Co_2$ it is
an elementary abelian 2-group of order $2^{23}$, and it is isomorphic to
the submodule
of index 2 stabilized by $Co_2$ in the Leech lattice mod 2. This
module contains a one-dimensional subspace invariant under the action
of $Co_2$ and the quotient over this subspace is also a representation
group (\cite[(5.7)]{IPS}). The representation groups of the $T$-geometries
for $M_{24}$
and $Co_1$ are elementary abelian of order $2^{11}$ resp. $2^{24}$ and
isomorphic to the 11-dimensional irreducible submodule
of the Golay co-code resp. to the Leech lattice mod 2
(\cite[(3.2),(5.8)]{IPS}). Also by \cite{IPS} the universal
representation group of the $P$-geometry for the Babymonster
sporadic simple group $BM$ is a non-split
extension $2\!\cdot\! BM$ and the universal representation group of the
$T$-geometry for the Monster $M$ is isomorphic to $M$. By an unpublished
result of A. Ivanov and S. Shpectorov there exist examples of
$c$-extensions with automorphism groups $(2\!\cdot\!
BM\! *\! 2\!\cdot\! BM)\! :\! 2$ and $(M\!\times\! M)\! :\! 2$, which
are related to the representation group in these cases as well.\par
The aim of the present paper is to show that under the assumption of ($*$) all
$c$-$T$-geometries arise in the context of the representation group and
that there are only three more
examples of $c$-$P$-geometries , which we will describe in Section
\ref{sec3}. More precisely we prove
\begin{satz}\label{satz1}
Let $G\le Aut(\Gamma )$ be a flag-transitive automorphism-group of a
$c$-$P$-geometry
$\Gamma$ of rank $n\ge 5$. Assume that the residue of
each flag of type $\{1,\ldots ,n-3\}$ is the $P$-geometry
for $M_{22}$.
\begin{enumerate}
\item[(a)] If $n=5$ and the residue of an element of type 1 is the $P$-geometry for $M_{23}$ then $G\cong M_{24}$.
\item[(b)] If $n=5$ and the residue of an element of type 1 is the
$P$-geometry for $Co_2$ then either $G\cong 2^{22}\! :\!Co_2$ resp. $2^{23}\! :\!Co_2$ and $O_2(G)$ considered as module for $G/O_2(G)$ is isomorphic to a section of the Leech lattice mod 2; or $G\cong Co_1$.
\item[(c)] If $n=6$ then either $G$ is isomorphic to one of the groups $BM\!\times\! BM$, $BM\wr {\bf Z}_2$,
$(2\!\cdot\! BM\! *\! 2\!\cdot\! BM)$, $(2\!\cdot\! BM)\wr {\bf Z}_2$,
or $G\cong M$.
\end{enumerate}
\end{satz}
\begin{satz}\label{satz2}
Let $G=Aut(\Gamma )$ be the automorphism-group of a $c$-$T$-geometry
$\Gamma$ of rank $n\ge 4$. Assume that the residue of
each flag of type $\{1,\ldots ,n-3\}$ is the tilde geometry
for $M_{24}$.
\begin{enumerate}
\item[(a)] If $n=4$ then
$G$ is the semidirect product of $M_{24}$ with the
(11-dimensional) Golay code module.
\item[(b)] If $n=5$ then $G$ is the semidirect product of $Co_1$ with
the Leech lattice mod 2.
\item[(c)] If $n=6$ then $G\cong M\times M$, or $M\wr {\bf Z}_2$.
\end{enumerate}
\end{satz}
We prove Theorems 1 and 2 in a series of lemmas in Section 4. In Section 2 we derive some general results about subgeometries of $\Gamma $, which apart from being useful for the determination of $Aut(\Gamma )$ also might be of their own interest. Section 3 contains some further preliminary results.\par
For the rest of the paper we fix the following notation:
$\Gamma $ is a $c$-$P$- or a $c$-$T$-geometry, $\Gamma ^{(i)}$ is the set of
objects
of $\Gamma $ of type $i$, $\{ \alpha _1,\ldots ,\alpha _n\} $ is a maximal
flag in $\Gamma $ with $\alpha _i\in \Gamma ^{(i)}$, $G\le Aut(\Gamma )$ is
a flag-transitive automorphism group of $\Gamma $,
$G_i=G_{\alpha _i}$ is the stabilizer of $\alpha _i$ in $G$, $K_i$ is the
kernel of the action of $G_i$ on the residue $res(\alpha _i)$ of $\alpha _i$
in $\Gamma $, $P_i=\bigcap_{j\neq
i}G_j$ are the so-called ``minimal parabolics'' and $B=P_1\cap \ldots \cap
P_n$ is the ``Borel subgroup''. Elements of type 1 will sometimes be
called ''points'', elements of type 2 ''lines'', and elements of type 3
''planes''. Further, for $x\in\Gamma ^{(i)}$, by
$res(x)^-$ we denote the geometry induced on the objects of types
$1,\ldots ,i-1$ in $res(x)$ and by $res(x)^+$ the geometry induced on the objects of types $i+1,\ldots ,n$.\par
By $2^n$ we usually denote an elementary abelian 2-group of order $2^n$, by $[2^n]$ any 2-group of order $2^n$, and by $2^{n_1+n_2+\ldots +n_k}H$ we mean the extension of a 2-group $P$ by a group $H$ such that $P$ has an $H$-invariant composition series $1=P_0\unlhd P_1\unlhd \ldots\unlhd P_k=P$ with $P_i/P_{i-1}\cong 2^{n_i}$. The rest of the notation is standard.
\section{Examples of $c$-$P$-geometries}\label{sec3}
\subsection{A geometry for $M_{24}$ with diagram}\label{exm24}
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This example is probably well known but we include it for completeness.
Let ${\cal S}={\cal S}(24,8,5)$ be the Steiner system for $G=M_{24}$
with underlying set $\Omega $ (for some properties of $\cal S$ see
e.g. \cite[chapters 6,7]{Asch}). Let ${\cal O}$ denote the set of octads and
${\cal T}$ the set of trios. We set
\begin{enumerate}
\item[--] $\Gamma^{(1)}=\Omega $,
\item[--] $\Gamma ^{(2)}=\{ \{ a_1,a_2\} |a_i\in\Omega , a_1\neq a_2\} $
\item[--] $\Gamma ^{(3)}=\{ \{ a_1,a_2,a_3,a_4\} |a_i\in \Omega $, $a_i\neq
a_j$ for $i\neq j\}$
\item[--] $\Gamma ^{(4)}=\{ (O_1,\{ O_2,O_3\})|\{ O_1,O_2,O_3\}\in{\cal T}\}$
\item[--] $\Gamma ^{(5)}={\cal O}$.
\end{enumerate}
Incidences between objects of types 1,2,3 are defined by inclusion. An
element $a\in
\Gamma ^{(1)}$ is incident to $(O_1,\{ O_2,O_3\})\in \Gamma ^{(4)}$ if
$a\in O_1$, and to $O\in \Gamma ^{(4)}$ if $a\not\in O$. Elements $x\in \Gamma
^{(2)}\cup \Gamma ^{(3)}$ and $y\in \Gamma ^{(4)}\cup \Gamma ^{(5)}$ are
incident if all elements of $x$ are incident to $y$. The elements of type 5
in $res(x)$ for $x=(O_1,\{ O_2,O_3\})\in \Gamma ^{(4)}$ are $O_2$ and $O_3$.
\subsection{A geometry for $Co_1$ with diagram}
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In order to define the geometry for $Co_1$ first we briefly recall the
definition of the P-geometry for $Co_2$ as it is given in \cite{Sh1}.\par
Let $\Lambda $ denote the Leech lattice with inner
product ( , ) and let $\Lambda _2=\{ v\in \Lambda , (v,v)=32\} $. Consider the
group $H=Co_2$ as the stabilizer in $Aut(\Lambda )$ of a fixed vector $v_0\in
\Lambda _2$. Let $\Sigma =\{ \{ v,-v\} |v\in \Lambda _2, (v_0,v)=0\} $ and
define a graph on $\Sigma $ in the way that $\sigma $ and $\tau
\in \Sigma $ are adjacent if $H_{\sigma}\cap H_{\tau }\cong 2^9.2^5\Sigma
_5$. A clique ${\cal C}$ in this graph is called closed if for each
$\{ v,-v\} , \{ w,-w\}\in {\cal C}$ there exists $\{ u,-u\}\in {\cal C}$ such
that $v_0+v+w+u\in 2\Lambda $. It is shown in \cite{Sh1} that each closed
clique is of size 1,3,7 or 15, and that, if we take as
objects of type $i$ the closed cliques of size $2^i-1$ and define incidence
by inclusion, then we get the P-geometry for $Co_2$. \par
Now we can define the $c$-$P$-geometry for $Co_1$: Let
\begin{enumerate}
\item[--]$\Gamma ^{(1)}=\{ \{ v,-v\} |v\in \Lambda _2\} .$
\end{enumerate}
For $x=\{ v,-v\}\in \Gamma ^{(1)}$ let $\Sigma _x=\{ \{ w,-w\}\in \Gamma
^{(1)}| (v,w)=0\} $ and consider $\Sigma _x$ as the graph just described. Let
\begin{enumerate}
\item[--]$\Gamma ^{(i)}=\{ \{ x\} \cup {\cal C}|x\in \Gamma ^{(1)}, {\cal
C}\subseteq \Sigma _x, |{\cal C}|=2^{i-1}-1$, ${\cal C}$ is a closed
clique in the graph on $\Sigma _x\} $, for $2\le i\le 5$,
\end{enumerate}
and define incidence by inclusion. Then it is
straightforward to show that $\Gamma =\bigcup_{i=1}^5\Gamma ^{(i)}$ is a
geometry with the desired diagram. Since $Co_1$ is transitive on
$\{ \{ v,-v\} |v\in \Lambda _2\} $, it is clear that $Aut(\Gamma )\cong Co_1$.
\subsection{A geometry for the Monster with diagram}
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The idea of the construction is quite similar to the previous example.
Recall that $2\cdot BM=C_M(z)$ for an involution
$z\in M$ of type $2A$ in the notation of \cite{A} and that the objects of
the P-geometry for $BM$ can be identified with certain conjugacy classes of
elementary abelian 2-groups of orders $2,4,8,16$ resp. $32$ of $BM$ (see
e.g. \cite{IPS}). Therefore, we
define $\Gamma $ in the following way:
\begin{enumerate}
\item[--] $\Gamma^{(1)}=2A$,
\item[--] $\Gamma ^{(i)}=\{ zU| z\in 2A$, $U\le C_M(z)$, $U$ is elementary
abelian of order $2^{i-1}$ and corresponds to an object of the P-geometry
for $C_M(z)/\langle z\rangle \cong BM\} $, for $2\le i\le 6$.
\end{enumerate}
Incidence is again defined by inclusion.\par
If $zU\in \Gamma ^{(i)}$ for some $i>1$ and $u\in U$, then $u$ is of type
$2B$ in $M$, i.e. $C_M(u)\cong 2^{1+24}Co_1$, and $\langle z,U\rangle \le
O_2(C_M(u))$. Since $O_2(C_M(u))$ is extraspecial and $N_M(U)\cap C_M(z)$ is
transitive on $U$, $N_M(\langle z, U\rangle )$ is transitive on $zU$. Now it
is straightforward to see that $\Gamma $ is a geometry of $c$-$P$-type. \\
\\
\section{Subgeometries}\label{sec1}
In this section $\Gamma $ always denotes a $c$-$P$-geometry of rank
$n\ge 4$. The reader might take it as an exercise to generalize
the results to any flag-transitive geometry
with diagram \Diagn{1}{2}{n-2}{n-1}{n}{X}, where
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\end{picture}
may be an arbitrary rank-2-geometry in which every object of the rightmost type is
incident to exactly three objects of the other type. Further one has to assume that the
stabilizer of an element $x\in \Gamma $ of type $i\ge 2$
induces the full affine linear group $2^{i-1}L_{i-1}(2)$ on $res(x)^-$. In
particular, we need the fact that $G_x$ acts 3-fold
transitively on the set of points and 2-fold transitively
on the set of partitions of $res(x)^-$ into classes of parallel lines.
(This is automatically true if $X=P^*$.)
\subsection{Shrinking}
In this subsection we define a new incidence
structure $\cal{S}$ related to $\Gamma $. In general, the structure $\cal S$ might not be connected (in fact,
for most of the $c$-$P$-geometries it is disconnected), but it is
not dificult to see that its connected components are
geometries with the same diagram as $\Gamma $ but of rank $n-1$. (Therefore,
we call this construction ``shrinking''.) \par
The structure ${\cal S}$ has the following sets of objects:
\begin{enumerate}
\item[--] ${\cal S}^{(1)}=\Gamma^{(2)}$,
\item[--] ${\cal S} ^{(i)}=\{ (x,P_x)| x\in \Gamma^{(i+1)}$, $P_x$ is a
partition of $res(x)^-$ into parallel lines$\}$,
\end{enumerate}
for $2\le i\le n-1$.
The elements $(x,P_x)\in {\cal S} ^{(i)}$, $(y,P_y)\in {\cal S} ^{(j)}$ with
$i,j>1$, are incident if $x$ and $y$ are incident in $\Gamma
$ and $P_x$ is compatible with $P_y$; $x\in {\cal S} ^{(1)}$ and $(y,P_y)\in
{\cal S} ^{(i)}$, $i>1$, are incident if $x$ is incident to $y$ in $\Gamma
$ and is part of the partition $P_y$ (recall that $x$ is a line).\par
\bigskip
In the following we derive some properties of $\cal S$. The first lemma is due to A. Pasini.
\begin{lem}\label{lem3} Let $\cal S$ be
disconnected. Then, for every $x\in \Gamma ^{(i)}$, $i\ge
3$, and two different partitions $P_1$,$P_2$ of $res(x)^-$ into parallel lines, the
objects $(x,P_1)$ and $(x,P_2)$ belong to different connected components of
$\cal S$.
\end{lem}
{\it Proof.}~~The relation ``giving rise to objects in the same connected
component'' is clearly an equivalence relation on the set of partitions of $res(x)^-$
into parallel lines and it is preserved by the action of $G_x$ on that
set. But as mentioned above that action is 2-transitive and therefore
primitive. So either we
have the statement or
all partitions give rise to objects in the same connected component of $\cal
S$. However, in the latter case the connectedness of $\Gamma $ forces $\cal
S$ to be connected. \hfill $\Box $\\
We can also describe the shrinking in terms of the
corresponding parabolic subgroups:
If $P_1,\ldots ,P_n$ are the minimal parabolics corresponding to
a fixed maximal flag of $\Gamma $ and $R_1,\ldots ,R_{n-1}$ the minimal
parabolics belonging to a certain maximal flag in a
suitable connected component $\Sigma $ of $\cal{S}$, then $R_i=\langle
P_1,P_{i+1}\rangle $ for $i\ge 2$ and $R_1=N_{\langle P_1,P_2\rangle }(P_1)
$. If $G_{\Sigma }$ denotes the stabilizer of $\Sigma $ in $G$,
then clearly $G_{\Sigma }=\langle R_1,\ldots ,R_{n-1}\rangle $ and $\cal S$
is connected iff $G=G_{\Sigma }$. For the Borel-subgroup
$B_{\Sigma }$ we have $B_{\Sigma }=R_1\cap\ldots\cap R_{n-1}=P_1$, hence
$|B_{\Sigma }:B|=2$.
\begin{lem}\label{lem2}
Let $G_{\Sigma }$ be the stabilizer of a connected component $\Sigma $ of
$\cal S$, $N\unlhd G_{\Sigma }$ the kernel of the action of $G_{\Sigma }$ on
$\Sigma $ and $H=G_{\Sigma }/N$. Suppose
\begin{enumerate}
\item[(a)] the kernel of the action of $H_p$ on the residue of $p$ in
$\Sigma $ is trivial for $p\in
\Sigma $ of type 1,
\item[(b)] $G_2/K_2\cong {\bf Z}_2\times G_{12}/K_2$.
\end{enumerate}
Let $Q_2\unlhd G_2$ such that $|Q_2:K_2|=2$ and $G_2/Q_2\cong
G_{12}/K_2$. Then $N=Q_2$.
\end{lem}
{\it Proof.}~~We identify $p\in \Sigma $ with $\alpha _2\in\Gamma
$ and $H_p$ with $G_2N/N$. Now $Q_2$ is trivial on the elements of types
$3,\ldots ,n$ of $res(\alpha _2)$ in $\Gamma $, hence it is trivial on
the residue of
$p$ in $\Sigma $. By assumption (a) $Q_2\le N$. \par
Let $F_{\Sigma }$ be a maximal flag in $\Sigma $. Then
$F_{\Sigma }$ is stabilized by $N$. By
construction, $F_{\Sigma }$ comes from a unique flag $F$ of cotype 1 in
$\Gamma $. So $N$ stabilizes $F$. Since $F$ is contained in exactly two
maximal flags, we have $|N:N\cap B|\le 2$. On the other hand,
our construction implies that $N\cap B\le K_2$. So the assertion follows
from $Q_2\le N$ and $|Q_2:K_2|=2$.\hfill $\Box $\\
\begin{cor}\label{cor1}
Suppose the assumptions of Lemma \ref{lem2}. Then ${\cal S}$ is
disconnected.
\end{cor}
{\it Proof.}~~ By Lemma \ref{lem2} $Q_2=N\unlhd G_{\Sigma }$. Since $Q_2\le
G_2$, therefore $Q_2$ must stabilize each
element of $\{ \alpha _2^{G_{\Sigma }}\}$. Hence $G_{\Sigma }\neq G$.\hfill
$\Box $\\
If $n\ge 5$, we can repeat the shrinking construction applying it to $\cal
S$. Then we get a new incidence structure, say $\cal T$, whose
connected components
are contained in the connected components of $\cal S$ and constitute
$c$-$P$-geometries of rank $n-2$.
For the set ${\cal T}^{(1)}$ of objects of $\cal T$ of type 1 we have by construction
$${\cal T}^{(1)}={\cal S}^{(2)}=\{ (x,P_x)| \hbox{$x\in \Gamma^{(3)}$, $P_x$ is a partition of $res(x)^-$ into parallel lines$\} $.} $$
Now, for each $x\in\Gamma ^{(3)}$, there are exactly three different
partitions $P_1,P_2,P_3$ of $res(x)^-$ into two parallel lines, and by Lemma
\ref{lem3} either $\cal S$ is connected or each pair $(x,P_i)$ belongs to a
different connected component of $\cal S$, hence also to a different
connected component of $\cal T$. Define a relation $\equiv $ on the set of
connected components of $\cal T$ by
$\Theta _1:\equiv \Theta _2$ if there exist $x\in \Gamma^{(3)}$ and
partitions $P_1,P_2$ of $res(x)^-$ into parallel lines such that $(x,P_i)$ belongs
to $\Sigma _i$ for $i=1,2$. Then the following holds.
\begin{lem}
Suppose $\cal S$ is disconnected and define the relation $\equiv$ as just
described.
If $\Theta _1\equiv \Theta _2$ then the elements of $\Theta _1^{(1)}$ and
$\Theta _2^{(1)}$ arise from the same set of elements from
$\Gamma^{(3)}$. Moreover,
$\equiv $ is an equivalence relation on the set of connected components of
$\cal T$, each equivalence class is of size three and, if $\{\Theta
_1,\Theta _2,\Theta _3\}$ is such an equivalence class, then the $\Theta _i$
are contained in different connected components of $\cal S$.
\end{lem}
{\it Proof.}~~ Obviously $\equiv$ is reflexive and symmetric. Moreover, if
the first statement of the lemma is shown, then it is clear that $\equiv $
is also transitive, hence an equivalence relation, and Lemma \ref{lem3}
together with the fact that, for any $x\in \Gamma ^{(3)}$, there are exactly
three partitions of $res(x)^-$ into parallel lines implies the rest of the
assertions. \par
Let $\Theta _1\equiv \Theta _2$ and $x\in
\Gamma^{(3)}$ such that $(x,P_i)\in \Theta _i$ for certain partitions
$P_1,P_2$ of $res(x)^-$. By $res(x,P_i)^{(2)}$ denote the set of elements of
type 2 in the residue of $(x,P_i)$ in $\Theta _i$. It follows from the
construction of $\cal T$ that
\begin{enumerate}
\item[$res$]$(x,P_i)^{(2)}=\{ (l,P_{S,l})|l\in {\cal S}^{(3)}$, $l$ is
incident to $(x,P_i)$ in ${\cal S}$, and $(x,P_i)$ is part of the
partition $P_{S,l}$ of the residue of $(l,P_{S,l})$ into parallel lines of
${\cal S}\} $.
\end{enumerate}
As $l\in {\cal S}^{(3)}$ and $l$ is incident to $(x,P_i)$, by
construction of $\cal S$ there is $l_0\in\Gamma ^{(4)}\cap res(x)$ and
a partition $P_0$ of $res(l_0)^-$ into parallel lines of $\Gamma $ such that
$l=(l_0,P_0)$ and $P_0$ is compatible with the partition $P_i$. In terms of $\Gamma $ then $x$ is a
plane of $l_0$ and $P_{S,l}$ induces a partition of $res(l_0)^-$ into two parallel
planes. If $(x',P_i')\in \Theta _i^{(1)}$ is the unique element incident to
$(l,P_{S,l})$ in $\Theta _i$ and different from $(x,P_i)$, then $x'$ is
also a plane of $res(l_0)^-$ and $(x',P_i')$ must be part of
the partition $P_{S,l}$. So $x'$ must be the
plane in $res(l_0)^-$ parallel to $x$. This means that
there is a bijection between the
set of points collinear to $(x,P_i)$ in $\Theta _i$ and the planes parallel
to $x$ in $\Gamma $. Since $\Theta _i$ is connected, this implies the
assertion. \hfill
$\Box $\\
\subsection{From $c$-$P$- to $T$-geometries}\label{sec12}
In this subsection we show how, under certain conditions, starting from a
$c$-$P$-geometry $\Gamma $ we can
construct a tilde geometry with the same automorphism group.\par
\bigskip
Let us first assume the rank of $\Gamma $ is 4 and
$G=Aut(\Gamma )\cong M_{24}$. We will identify $\Gamma $
with the geometry from \cite[Example 2]{AC}. So $\Gamma $ has the
following sets of objects (where $\Omega $, $\cal T$, and $\cal O$ are as in
Example \ref{exm24} the points, trios and octads of the Steiner system):
\begin{itemize}
\item[--] $\Gamma^{(1)}=\{ \{ p,q\} |p,q\in\Omega , p\neq q\} $
\item[--] $\Gamma^{(2)}=\{ \{ a,b\} |a,b\in \Gamma _1, a\cap b=\emptyset \}$
\item[--] $\Gamma^{(3)}=\{ (O_1,P,\{ O_2,O_3\} )| O_i\in{\cal O}, \{
O_1,O_2,O_3\}\in{\cal T}$, $P$ is a partition of $O_1$ into four parallel
lines in the affine geometry on $\Omega\setminus O_2\} $
\item[--] $\Gamma^{(4)}=\{ (O,P) |O\in{\cal O}$, $P$ is a partition of
$\Omega\setminus O$ into parallel lines $\}$.
\end{itemize}
(Recall that for $O\in {\cal O}$ the complement
$\Omega\setminus O$ bears the structure of an affine space over $GF(2)$ and $G_O\cong 2^4A_8$ acts as affine group $AGL(4,2)$ on $\Omega\setminus O$.)\par
Let $\Sigma $ be a
connected component of the structure
$\cal S$ obtained from $\Gamma $ by the shrinking construction described in
the previous section. Then $\Sigma $ has the diagram
\begin{picture}(25,5)
\put(2,1){\makebox(0,0){$\circ$}}
\put(12,1){\makebox(0,0){$\circ$}}
\put(22,1){\makebox(0,0){$\circ$}}
\put(2.5,1){\line(1,0){8.7}}
\put(12.5,1){\line(1,0){8.7}}
\put(7,2.5){c}
\put(17,2.5){$\scriptsize P^*$}
\put(1,2.5){\scriptsize {1}}
\put(11,2.5){\scriptsize{2}}
\put(21,2.5){\scriptsize{3}}
\end{picture} .
We have $G_2\le G_{\Sigma }$ and $G_2\cong 2^{4+1}({\bf Z}_2\times\Sigma _5)$, and Lemma
\ref{lem2} resp. Corollary \ref{cor1} yield that $\Sigma $ is disconnected
and that $O_2(G_2)$ is the kernel of the action of $G_{\Sigma }$ on $\Sigma
$. In particular, $O_2(G_2)\unlhd G_{\Sigma}$, and the list of maximal
subgroups of $M_{24}$ in \cite{A} shows that $G_{\Sigma }\le G_S$, where
$G_S\cong 2^63\Sigma _6$ and $G_S$ is the stabilizer of a sextet $S$ in the
Steiner system. From the structure of $G_S$ we see
that either $G_{\Sigma }=G_S$ or
$G_{\Sigma }\cong 2^6({\bf Z}_3\times \Sigma _5)$. But in the latter
$|G_{\Sigma }:G_2|=3$, so $\Sigma $ would just contain 3 points, which
is not true (consider the residue of an element of type 3). Hence $G_{\Sigma }=G_S\cong 2^63\Sigma _6$. \par
Let $T^{(1)}$ be the set of connected components of $\cal S$, $T^{(2)}$ the
set of triples of elements from $\Gamma ^{(3)}$ of the shape
$$\{ (O_1,P_1,\{ O_2,O_3\} ), (O_2,P_2,\{ O_1,O_3\} ),
(O_3,P_3,\{ O_1,O_2\} )\} $$
such that $P_i\cup
P_j$ is a partition into parallel lines of the affine space $\Omega
\setminus O_k$ for all
choices of $\{ i,j,k\} =\{ 1,2,3\} $, and let $T^{(3)}=\Gamma ^{(4)}$. Set
$T=T^{(1)}\cup T^{(2)}\cup T^{(3)}$ and define an incidence relation on $T$
in the way that $\Sigma \in T^{(1)}$ and $x\in T^{(2)}\cup T^{(3)}$ are
incident if $(x,P_x)$ (resp. $(y,P_y)$ for all $y\in x$) belong to $\Sigma $,
where $P_x$, $P_y$ are suitable partitions of $res(x)^-$ resp. $res(y)^-$
into parallel lines of $\Gamma $. The elements in $T^{(3)}$ incident
to a typical element
$$\{ (O_1,P_1,\{ O_2,O_3\} ), (O_2,P_2,\{ O_1,O_3\} ),
(O_3,P_3,\{ O_1,O_2\} )\} \in T^{(2)}$$
are the three elements $(O_i,P_j\cup
P_k)$, $\{ i,j,k\} =\{ 1,2,3\} $.\par
If $x=(O,P)\in T^{(3)}$ then $G_x\le G_O\cong 2^4A_8$. Since $G_x$ stabilizes
the partition $P$ we see $G_x\cong 2^{1+6}L_3(2)$. If $x\in T^{(2)}$, $x$ as above, then
$G_x\le G_t$ where $G_t$ is the stabilizer of the trio $t=\{ O_1,O_2,O_3\}$ and $G_t\cong [2^6](\Sigma _3\times L_3(2))$. The $\Sigma _3$-factor permutes the three octads in $t$, so it will stabilize $x$ as a set. The $L_3(2)$-factor can be viewed as the stabilizer of a partition of $\Omega\setminus O_i$ into two parallel planes for each of the $O_i$. Since $G_x$ also
stabilizes the partitions $P_j\cup P_k$ we deduce $G_x\cong
[2^8](\Sigma _3\times \Sigma _3)$. So we have
shown that $G_x$ is of the right shape for all $x\in T$. We leave it to the
reader to convince himself that the incidence relation is defined in the
right way to get a model of the tilde geometry for $M_{24}$.\par
\bigskip
Now let $\Gamma $ be of arbitrary rank $n\ge 5$. Apply the shrinking
construction $n-3$ times and call the obtained incidence structures ${\cal
S}_1$,
${\cal S}_2$, \ldots ,${\cal S}_{n-3}$. Suppose that ${\cal S}_1$ is
disconnected and that the connected components of ${\cal S}_{i+1}$ are not
equal to those of ${\cal S}_i$, $i=1,\ldots
,n-4$. Then, similarly as in section 2.1, for each $i\ge 2$, we can define an equivalence relation on the
set of connected components of ${\cal S}_i$, whose equivalence classes
correspond to the set of partitions into two parallel hyperplanes of an
$i$-dimensional affine space over $GF(2)$. Hence they are of size
$2^i-1$. Assume the connected components of ${\cal
S}_{n-4}$ are the $c$-$P$-geometries for $M_{24}$. Let
\begin{enumerate}
\item[--] $T^{(1)}=$ connected components of ${\cal S}_1$,
\item[--] $T^{(i)}=$ equivalence classes of connected components of ${\cal
S}_i$, for $2\le i\le n-3$,
\item[--] $T^{(n-2)}=$ triples of elements of $\Gamma ^{(n-1)}$ which, for
suitable partitions, are contained in a common connected component of
${\cal S}_{n-2}$ and there form a triple which corresponds to an element
of type 2 of the $T$-geometry for $M_{24}$,
\item[--] $T^{(n-1)}=\Gamma ^{(n)}$
\end{enumerate}
and $T$ be the geometry on $T^{(1)}\cup\ldots \cup T^{(n-1)}$ with the following incidene relation:\par
Two equivalence classes $x\in T^{(i)}$, $y\in T^{(j)}$, $1