\documentstyle[a4,12pt]{article}
\title{The geometry far from a residue}
\date{December 13, 1996}
\author{R. J. Blok \& A. E. Brouwer}
\begin{document}
\maketitle

\newtheorem{Theorem} {Theorem} [section]
\newtheorem{Proposition} [Theorem] {Proposition}
\newtheorem{Corollary} [Theorem] {Corollary}
\newtheorem{Lemma} [Theorem] {Lemma}
\newcommand{\Proof}{ \noindent{\bf Proof:}\quad }
\newcommand{\qed  }{\hfill$\Box$\medskip}
\newcommand{\Ff}{{\bf F}}
\newcommand{\Cham}{{\rm Cham}}
\newcommand{\typ}{{\rm typ \,}}
\newcommand{\op}{{\rm op}}
\newcommand{\proj}{{\rm proj}}
\newcommand{\Far}{{\rm Far}}
\newcommand{\Stab}{{\rm Stab}}

\begin{abstract}
We show that in general the subgeometry of a building of spherical type
induced by all objects in general position with respect to a given residue
is a residually connected geometry with a Buekenhout-Tits diagram resembling the 
original diagram, but with certain strokes replaced by the corresponding
`affine' strokes.
The exceptions (where connectedness fails) are investigated in some detail.
\end{abstract}

\section{Introduction}
Given a geometry $\Gamma$, and some substructure $S$, and a suitable notion
of `far from', we can study the subgeometry $\Delta$ of $\Gamma$ consisting
of objects far from $S$. We show that $\Delta$ has a Buekenhout-Tits
diagram related to that of $\Gamma$, and study the question whether
$\Delta$ will be connected.

For the geometry $\Gamma$ we shall take a building of spherical type.
We focus on the following three cases.

(i) Fix a chamber or residue $R$ of $\Gamma$,
and look at $\Far(R)$, the collection of all chambers opposite to $R$
(or, equivalently, at maximal distance from $R$), with the induced adjacency.

(ii) Fix a type $i$ in the diagram of $\Gamma$, and look at the points
at maximal distance from a given point in the corresponding point-line
geometry.

(iii) Fix a type $i$ in the diagram of $\Gamma$, and look at the
complement of a geometric hyperplane in the corresponding point-line
geometry.

The special case of buildings of rank 2 was treated in Brouwer \cite{B93}
using eigenvalue arguments. Roughly speaking, the result is that the
subgeometries considered are always connected, except in the cases
$B_2(2)$, $G_2(2)$, $G_2(3)$, ${}^2F_4(2)$. For a precise statement,
see Section \ref{ranktwo}.

In the present note we consider the higher rank case, find that it can be
reduced to the rank 2 case, and hence are left only with the geometries
that have $B_2(2)$ residues, namely $Sp(2n,2)$ and $F_4(2)$.

We find that far from a chamber the $Sp(2n,2)$ geometry has $2^{n-1}$
connected components, while in $F_4(2)$ there are 4 connected components.

In the interests of brevity, the arguments will be somewhat sketchy
in places. We hope to write down a fuller treatment elsewhere.

\section{The rank 2 case}
\label{ranktwo}
The results in the finite rank 2 case are as follows:

\begin{Theorem} {\rm (Brouwer \cite{B93})}
Let $(X, {\cal L} )$ be a thick finite generalized $n$-gon
of order $(s,t)$. Then
\begin{enumerate}
\item[{\rm (i)}]
The subgeometry induced by the points and lines in general position w.r.t.
a given point $x$ (or, indeed, the complement of
an arbitrary geometric hyperplane in $(X, {\cal L} )$)
is connected, except possibly
in the cases $(n,s,t) = (6,2,2)$, $(8,2,4)$.
\item[{\rm (ii)}]
The subgeometry induced by the points and lines in general position w.r.t.
a given flag $(x,L)$ is connected, except possibly
in the cases $(n,s,t) = (4,2,2)$, $(6,2,2)$, $(6,3,3)$,
$(8,2,4)$, $(8,4,2)$.
\item[{\rm (iii)}]
For the stated possibly exceptional parameter sets actual
exceptions do occur: in case (i) for $G_2(2)$ and ${}^2F_4(2)$,
and in case (ii) also for $B_2(2)$ and $G_2(3)$.
\end{enumerate}
\end{Theorem}

{\bf Problem} Show a direct connection with the fact that
$G_2(2)$, ${}^2F_4(2)$ and ${}^2B_2(2)$, ${}^2G_2(3)$ are not simple.

\medskip
For infinite generalized $n$-gons with $n \le 4$ the subgeometries
considered are all connected (\cite{B93}). On the other hand,
for $n \ge 5$ one may use free constructions to obtain generalized $n$-gons
such that the subgeometry on the points far from a given point is disconnected.
(Maybe this remark is due to P. Abramenko?)
For the Moufang $n$-gons Abramenko checked and found them connected
far from a chamber (according to \cite{MR}, p.73).

\section{The higher rank case}

If $A$ and $B$ are two sets of chambers, then let
$d(A,B) := \min(d(a,b) \mid a \in A, b \in B)$.

\begin{Theorem} {\rm (cf. M\"uhlherr \& Ronan \cite{MR}, Theorem 1.5)}
\label{connthm}
Let $\Gamma$ be a spherical building of rank at least 3.
If $R$ is a residue such that in every rank $2$ residue $X$
the subgeometry $\Far_X (\proj_X (R))$ is connected, then
in $\Gamma$ the subgeometry $\Far_{\Gamma}(R)$ is connected.
\end{Theorem}

(One may formulate a version that is also valid in the non-spherical
case: then the statement is that if in $X$ any two chambers can be
joined by a gallery, such that no chamber in this gallery is closer
to $\proj_X (R)$ than its extremities, then in $\Gamma$ any two
chambers can be joined by a gallery, such that no chamber in this
gallery is closer to $R$ than its extremities.)

We shall see below that the converse holds as well: if
some rank $2$ residue $X$ has a disconnected $\Far_{X}(\proj_X(R))$,
then also $\Far_{\Gamma} (R)$ is disconnected.


\section{The symplectic case}

\begin{Theorem}
Fix a chamber $c$ in the $Sp(2n,2)$ building, where $n > 0$,
and consider the geometry $\Delta$ of objects far from $c$.
Then $\Delta$ has $2^{n-1}$ connected components.
\end{Theorem}

There are various ways to understand the occurrence of components
here. Maybe the best way to describe these components is as
corresponding to $O^+(2n,2)$ subgeometries of the $Sp(2n,2)$ geometry.

Fix an $Sp(2n,2)$ geometry and an $O^+(2n,2)$ subgeometry.
In the $Sp(2n,2)$ chamber system $C$, an $O^+(2n,2)$ chamber
corresponds to a pair of $n$-adjacent chambers $(c',c'')$
(for the usual labeling of the diagram), two chambers $(c',c'')$
and $(d',d'')$ being $(n-1)$- (or $n$-) adjacent, when
$c'\ne d'$ but $c''=d''$ (or $c''\ne d''$ but $c'=d'$).
Let $D$ be the collection of chambers in $C$ occurring in such pairs.
Let $\{c,c',c''\}$ be an $n$-panel in $C$, with $c \in C \setminus D$,
$c',c'' \in D$. Then a chamber $d' \in D$ is far from $c$
if and only if it occurs in an $O^+(2n,2)$ chamber $(d',d'')$
opposite to $(c',c'')$. Since the projection of $c$ into the
$n$-panel $\{d,d',d''\}$ is not far from $c$, we see that
$n$-edges in $\Far(c)$ remain in $D$. Since it is clear that
$D$ is closed for $j$-edges, $j < n$, we see that any connected
component of $\Far(c)$ meeting $D$ is contained in $D$.
But $D \cap \Far(c)$ is connected (for example because the
$D_n$ diagram has no $B_2$ subdiagrams), so is a connected component.

Let us redo the above in geometric terms.
Let `$i$-object' stand for `totally isotropic (t.i.) $i$-space'.
Two chambers $c$ and $c'$ in an $Sp(2n,q)$ geometry are far
(opposite) if and only if whenever $A$ and $A'$ are $i$-objects
in $c$ and $c'$, respectively, we have $A^\perp \cap A' = 0$.

\begin{Proposition}
Fix an $n$-object $U$ with hyperplane ($(n-1)$-object) $H$ in the $Sp(2n,2)$
geometry. Let $F$ be the flag $(H,U)$.
The connected component of the geometry far from $F$ containing a given
$n$-object $V$ is the geometry far from $H$ in the $O^+(2n,2)$ geometry
defined by the quadric $Q$ that defines the given symplectic form,
for which $H$ and $V$ are totally singular, but $U$ is not.
(That is, $H = Q \cap U$.)
\end{Proposition}

The conditions that $Q$ defines the given symplectic form,
and vanishes identically on $H$ and $V$ but not on $U$,
and that $Q$ defines a hyperbolic quadric, indeed determine $Q$ uniquely.
If we choose a basis, then $Q$ is determined by the symplectic form,
except for its diagonal elements. Requiring that $Q$ vanishes on $H$
determines $n-1$ diagonal elements; one more is fixed since $Q$ must not
vanish on $U$; one more is fixed since $Q$ must be hyperbolic, so that
$H^{\perp} / H$ is a hyperbolic line; finally there are $n-1$ diagonal
elements to choose freely, giving $2^{n-1}$ connected components.

\section{The stabilizer of a component}

Let $\Gamma$ be a spherical Moufang building.
Fix an apartment $\Sigma$ with root system $\Phi$ and
a chamber $c$ of $\Sigma$.
Let $G$ be the group of automorphisms generated by the root
groups $U_{\alpha}$ $(\alpha \in \Phi)$.
Then, the pair $(B := \Stab_G(c), N := \Stab_G(\Sigma))$
is a $(B,N)$-pair for $G$ yielding the building $\Gamma$.\\
For a residue $R$ on $c$, define the following sets of roots:
$$\Phi(R,i) := \{ \alpha \in \Phi \mid c \in \alpha,~ d(R,-\alpha) = i \}$$
($i \ge 0$), and
$$\Phi(R) := \Phi(R,0) .$$
Put $H := B \cap N$, and consider the following subgroups of $G$:
$$\begin{array}{ll}
L(R) & = \langle H,U_{\alpha} \mid \alpha \in \Phi(R) \rangle \\
U(R) & = \langle U_{\alpha} \mid \alpha \in \Phi(R,i),~ i > 0 \rangle \\
P(R) & = U(R) . L(R) \\
U(R,1) & = \langle U_{\alpha} \mid \alpha \in \Phi(R,1) \rangle \\
P(R,1) & = U(R,1) . L(R,1)
\end{array}$$
The subgroup $P(R)$ is just the ordinary parabolic subgroup
stabilizing $R$ with Levi component $L(R)$ and unipotent radical
$U(R)$. The definition of $P(R,1)$ is justified by the
following theorem:

\begin{Theorem}\label{stabi}
The subgroup $P(R)$ of $G$ is the full stabilizer of $\Far_{\Gamma}(R)$
in $G$, and it acts regularly on the chambers of $\Far_{\Gamma}(R)$.
The subgroup $P(R,1)$ of $P(R)$ is the full stabilizer in $G$
of the connected component of $\Far_{\Gamma}(R)$ containing
$\op_{\Sigma}(R)$,
and it acts regularly on the chambers of this component.
In particular, $\Far_{\Gamma}(R)$ is connected if and only if
$P(R,1) = P(R)$.
\end{Theorem}
\Proof
The subgroup $L(R)$ acts regularly (i.e., sharply 1-transitively)
on the chambers of $R$ and on those of $\op_{\Sigma}(R)$.
The subgroup $U$ acts regularly on the set of residues opposite to $R$.
This implies that $P(R)$ acts regularly on the chambers of
$\Far_{\Gamma}(R)$.

Let $C$ be the connected component of $\Far_{\Gamma}(R)$ containing
$c' = \op_{\Sigma} (c)$.
Let $u \in U_{\alpha}$, $u \ne 1$.
If $\alpha \in \Phi(R)$, then $u c' \in \op_{\Sigma} (R)$
and if $\alpha \in \Phi(R,1)$, then there is a panel $\pi$
which determines $\alpha$ and which has precisely one chamber
$r \in R$. This chamber has at least two opposite chambers
in $\op_{\Sigma}(\pi)$, namely $r' := \op_{\Sigma}(r)$ and
$u r'$, and hence $u \, \op_{\Sigma} (R) \subseteq C$.
Thus, $P(R,1)$ preserves $C$.

Let $c' = r_0, ..., r_l = r$ be a gallery in $C$.
We show by induction on $l$ that there exists a unique
element $u \in P(R,1)$ which sends $c'$ to $r$.
For $l = 0$ this is clear.
Suppose $v \in P(R,1)$ sends $c'$ to $r_{l-1}$ and consider
the panel $\pi$ on $c'$ and $r' = v^{-1} r$.
If $r' \in \op_{\Sigma} (R)$, then there exists an element
$u' \in L(R)$ with $u' c' = r'$.
If $r' \notin \op_{\Sigma} (R)$, then $\op_{\Sigma}(\pi)$
only has the chamber $c$ in common with $R$, and so it determines
a root $\alpha$ in $\Phi(R,1)$.
The chamber $\proj_{\pi}(c)$ lies on $\alpha$ and since
$R \cap \Sigma \subseteq \alpha$, it lies outside $\Far(R)$.
It follows that both $c'$ and $r'$ lie opposite $c$, and so there
again exists an element $u' \in U_{\alpha}$ with $u' c' = r'$.
In both cases the element $u = u' v \in P(R,1)$ sends $c'$ to
$r$, and, as $P(R,1) \subseteq P(R)$, it is uniquely determined
by this property. \qed

Define
$$\Phi_S(R,i) := \Phi(S) \cap \Phi(R \cap S, i)$$
and
$$\Phi_S(R) := \Phi_S(R,0).$$

\begin{Lemma}
For any two residues $R$ and $S$ on $c$ we have

(i) $\Phi_S(R) = \Phi(S \cap R) = \Phi(S) \cap \Phi(R)$, and

(ii)  $\Phi_S(R,i) = \Phi(S) \cap \Phi(R,i)$.
\end{Lemma}

Let $L_S(R)$, $U_S(R)$, $P_S(R)$, $U_S(R,1)$ and $P_S(R,1)$
be the groups corresponding to the sets $\Phi_S(R)$ and $\Phi_S(R,i)$.
Then these groups are subgroups of $L(S)$, and using the previous lemma, one can prove the 

\begin{Proposition}\label{relgroups}
For any two residues $R$ and $S$ on $c$ we have

(i) $L_S(R) = L(S \cap R) = L(S) \cap L(R)$,

(ii) $U_S(R) = L(S) \cap U(R)$,

(iii) $P_S(R) = L(S) \cap P(R)$,

(iv) $U_S(R,1) = L(S) \cap U(R,1)$ and

(v) $P_S(R,1) = L(S) \cap P(R,1)$.
\end{Proposition}
The following corollary, which follows easily from Theorem \ref{stabi} and Proposition \ref{relgroups}, is the converse to 
 Theorem \ref{connthm}.

\begin{Corollary}
Let $R$ and $S$ be residues on a common chamber $c$.
If $\Far_S(S \cap R)$ is disconnected, then so is $\Far_{\Gamma}(R)$.
\end{Corollary}

\begin{Corollary}
Let $\Gamma$ be a spherical building of rank at least 3,
with a naturally labeled diagram $M$, and let $R$ be a $J$-residue.
Then $\Far_{\Gamma}(R)$ is disconnected when either

(i) $M = C_n$, $\Gamma$ is defined over the field $\Ff_2$ and
$J \cap \{n-1,n\} = \emptyset$, or

(ii) $M = F_4$, $\Gamma$ is defined over the field $\Ff_2$ and
$J \cap \{2,3\} = \emptyset$.
\end{Corollary}

{}From the theory above, it is very easy to reprove that in
the symplectic case there are $2^{n-1}$ connected components,
and after some fiddling it also follows that $F_4(2)$ has 4
connected components far from a chamber.

\section{The Buekenhout-Tits diagram of the geo\-metry far from a residue}

In this section we show that the  subgeometry of a building of
spherical type obtained by fixing a flag $F$ and taking all
objects `far away from' (or `in general position w.r.t.') $F$
has a Buekenhout-Tits diagram (cf. \cite{Bue81}).

\begin{Theorem}
Let $\Gamma$ be (the geometry of) a building of spherical type,
and fix a flag $F$. The subgeometry $\Far_{\Gamma}(F)$ of $\Gamma$
consisting of all elements of $\Gamma$ far away from
(that is, opposite to) $F$ (with inherited type and incidence)
is a geometry with a Buekenhout-Tits diagram obtained from that of $\Gamma$
by adding arrows pointing towards the nodes in $\op(\typ F)$.
\end{Theorem}

As always in Buekenhout geometry, the theorem is a claim for
geometries of rank at least 3, and is a definition of the strokes
involved for geometries of rank 2. Let us repeat this definition
explicitly.

The three diagrams
\smallskip
% \input far-pict.sty
% \usepackage{graphics}
\newsavebox{\node}
\savebox{\node}(8,8){\put(0,0){\makebox(8,8)[bl]{\put(4,4){\circle{8}}}}}
\newsavebox{\Gtgon}
\newsavebox{\lafGtgon}
\newsavebox{\rafGtgon}
\newsavebox{\bafGtgon}
\newsavebox{\Gmgon}
\newsavebox{\lafGmgon}
\newsavebox{\rafGmgon}
\newsavebox{\bafGmgon}
%----
\savebox{\Gtgon}(40,20){%
\put(0,0){\makebox(40,20)[bl]{%
\put(2,10){\usebox{\node}}
\put(10,14){\line(1,0){20}}
\put(30,10){\usebox{\node}}
}}}
%----
\savebox{\lafGtgon}(40,12){%
\put(0,0){\makebox(40,12)[bl]{%
\put(2,10){\usebox{\node}}
\put(30,14){\vector(-1,0){20}}
\put(30,10){\usebox{\node}}
}}}
%----
\savebox{\rafGtgon}(40,12){%
\put(0,0){\makebox(40,12)[bl]{%
\put(2,10){\usebox{\node}}
\put(10,14){\vector(1,0){20}}
\put(30,10){\usebox{\node}}
}}}
%----
\savebox{\bafGtgon}(40,12){%
\put(0,0){\makebox(40,12)[bl]{%
\put(2,10){\usebox{\node}}
\put(20,14){\vector(-1,0){10}}
\put(20,14){\vector(1,0){10}}
\put(30,10){\usebox{\node}}
}}}
%----
\savebox{\Gmgon}(40,20){%
\put(0,0){\makebox(40,20)[bl]{%
\put(2,10){\usebox{\node}}
\put(10,14){\line(1,0){20}}
\put(20,18){\makebox(0,0){${\scriptstyle m_{ij}}$}}
\put(6,4){\makebox(0,0){${\scriptstyle i}$}}
\put(34,4){\makebox(0,0){${\scriptstyle j}$}}
\put(30,10){\usebox{\node}}
}}}
%----
\savebox{\lafGmgon}(40,12){%
\put(0,0){\makebox(40,12)[bl]{%
\put(2,10){\usebox{\node}}
\put(30,14){\vector(-1,0){20}}
\put(20,18){\makebox(0,0){${\scriptstyle m_{ij}}$}}
\put(6,4){\makebox(0,0){${\scriptstyle i}$}}
\put(34,4){\makebox(0,0){${\scriptstyle j}$}}
\put(30,10){\usebox{\node}}
}}}
%----
\savebox{\rafGmgon}(40,12){%
\put(0,0){\makebox(40,12)[bl]{%
\put(2,10){\usebox{\node}}
\put(10,14){\vector(1,0){20}}
\put(20,18){\makebox(0,0){${\scriptstyle m_{ij}}$}}
\put(6,4){\makebox(0,0){${\scriptstyle i}$}}
\put(34,4){\makebox(0,0){${\scriptstyle j}$}}
\put(30,10){\usebox{\node}}
}}}
%----
\savebox{\bafGmgon}(40,12){%
\put(0,0){\makebox(40,12)[bl]{%
\put(2,10){\usebox{\node}}
\put(20,14){\vector(-1,0){10}}
\put(20,14){\vector(1,0){10}}
\put(20,18){\makebox(0,0){${\scriptstyle m_{ij}}$}}
\put(6,4){\makebox(0,0){${\scriptstyle i}$}}
\put(34,4){\makebox(0,0){${\scriptstyle j}$}}
\put(30,10){\usebox{\node}}
}}}
%----
\newsavebox{\afdaf}
\savebox{\afdaf}(210,20){%
\put(0,0){\makebox(210,20)[bl]{%
\put(2,10){\usebox{\node}}
\put(60,0){\usebox{\lafGmgon}}
\put(110,0){\usebox{\rafGmgon}}
\put(160,0){\usebox{\bafGmgon}}
}}}
%----

\begin{center}
\usebox{\Gmgon} ~ \usebox{\rafGmgon} ~ \usebox{\bafGmgon}
\end{center}
denote, respectively, (i) the class of all generalized $m_{ij}$-gons,
(ii) the class of all subgeometries of a generalized $m_{ij}$-gon found by
taking all objects in the incidence graph at distance $m_{ij}-1$ or $m_{ij}$
from a given $i$-object when $m_{ij}$ is odd, or from a given $j$-object
when $m_{ij}$ is even, and (iii)
the class of all subgeometries of a generalized $m_{ij}$-gon found
by taking all objects in flags at distance $m_{ij}$ from a given flag
in the flag graph (the line graph of the incidence graph).
As customary, we delete all (arrowed) edges labeled 2
(with or without arrows these all represent generalized digons),
we delete labels 3, and use double edges instead of edges labeled 4.

Thus, the diagrams
\begin{center}
\usebox{\Gtgon} ~ \usebox{\lafGtgon} ~ \usebox{\rafGtgon} ~ \usebox{\bafGtgon}
\end{center}
represent the classes of projective planes, affine planes,
dual affine planes and biaffine planes, respectively.
Instead of using arrows, one usually uses labels Af and Af* for
affine and dual affine planes.

The above theorem was inspired by a question of S. Shpectorov,
who asked for the classification of geometries with diagram
\begin{center}{\scriptsize
$$\begin{picture}(120,40)(0,-30)
\put(116.568,0){\circle{6}}
\put(88.284,0){\circle{6}}
\put(60,0){\circle{6}}
\put(20,0){\circle{6}}
\put(0,20){\circle{6}}
\put(0,-20){\circle{6}}
\put(91.284,0){\line(1,0){22.284}}
\put(63,0){\line(1,0){22.284}}
\put(17.879,2.121){\line(-1,1){15.757}}
\put(17.879,-2.121){\line(-1,-1){15.757}}
\put(23,0){\makebox(34,0){\dotfill}}
\put(14,14){\makebox(0,0){Af}}
\end{picture}
$$}\end{center}

Before proving the theorem, let us define the terms more precisely.
Two chambers $c,c'$ of a spherical building are called {\em opposite}
when $\delta(c,c') = w_0$. Since $w_0 = w_0^{-1}$, this is a symmetric relation.
Two residues are called opposite when they contain opposite chambers.
{\em Far} is synonymous with opposite.

A Coxeter chamber system $W$ becomes a building with the distance function
given by $\delta(x,y) = x^{-1}y$, and we see that each object has a unique
opposite.
Let $\op_W$ be the map defined on the Coxeter building $W$
by $x \mapsto xw_0$. This is an involution sending residues of type $J$
to residues of type $\op_I J$.

\begin{Lemma}
\label{opispo}
Let $W$ be a Coxeter building of spherical type. Then
for any two residues $R$ and $S$ we have
$\op_R(\proj_R(S)) = \proj_R(\op_W(S))$.
\end{Lemma}
\Proof
Since we defined $\op_R$ and $\proj_R$ elementwise, it suffices
to show $\op_R(\proj_R(x)) = \proj_R(\op_W(x))$ for a chamber $x \in W$.
But both projection and opposite are invariant under left multiplication,
so we may assume that $R = W_J$ is a subgroup of $W$.
Now $\proj_R(x) = p$ is the element of $R$ such that $p^{-1}x$ is the
unique shortest coset representative of the right coset $Rx$.
So we have to show that if $a$ is the shortest element of $Ra$, i.e.,
is left $R$-reduced, then $w_0(R) a w_0$ is shortest in $Raw_0$.
Or again, that $w_0(R) a$ is longest in $Ra$. But that holds,
since $l(ra) = l(r) + l(a)$ for all $r \in R$. \qed

Our first concern is to show that $\Delta := \Far_{\Gamma}(F)$
is a geometry.
Let $C$ be the chamber system of $\Gamma$, and consider the
collection $\cal F$ of all residues in $C$ (flags in $\Gamma$)
far from $F$.
We want to show that $\cal F$ is the flag complex of $\Delta$.
Now $\cal F$ will be a flag complex if any three pairwise incident
residues in $\cal F$ have a common chamber in $\cal F$.
But this follows immediately from the lemma below.

\begin{Lemma}
Let $R$ and $S$ be incident residues belonging to $\cal F$.
Then $R \cap S$ also belongs to $\cal F$.
\end{Lemma}
\Proof
Let $A$ be an apartment incident with $F$ and $R \cap S$.
Then in $A$ the three residues $\op F$ and $R$ and $S$ are
pairwise incident, and hence have a common element. \qed

\begin{Lemma}
Let $R$ and $S$ be opposite.
Then the set of residues meeting $\Far_{\Gamma}(S)$ and
contained in $R$ equals $\Far_R(\proj_R(S))$.
In particular, every object in $R$ belongs to $\Far_{\Gamma}(S)$
if and only if $\proj_R(S) = R$, that is, if and only if
$\op(\typ S) \subseteq \typ R$.
\end{Lemma}
\Proof
It suffices to prove this for a Coxeter complex $\Sigma$
instead of $\Gamma$ -- then the statements about $\Gamma$
follow by taking the union over all $\Sigma$ containing both
$R$ and $S$. By Lemma \ref{opispo} we are done.
\qed

\Proof (of the theorem).
Follows directly from the lemma above.
\qed

\section{Geometric hyperplanes and far subgeometries}
Often, geometric hyperplanes generalize subgeometries far from a point.
In particular, when one classifies the geometries having one of the diagrams
found in the previous section, then often the result one finds is that
they are complements of a geometric hyperplane in a building.

And indeed, the complement of a hyperplane in the point-line geometry
for node $i$ has diagram obtained from the original diagram by
inserting arrows pointing into $i$.

Suppose we have a building of spherical type, select a type $i$ to be
the point type, and consider the corresponding point-line geometry
(where the lines are the flags of cotype $i$).
If $P$ is a point, then does `not far from $P$' define a geometric hyperplane?

\begin{Proposition}
(i) The set of points of a point-line geometry as above, not far from
a given point $P$ is a geometric hyperplane if and only if
conjugation by $w_0$ does not move node $i$.
(ii) The set of points of a point-line geometry as above, not far from
a given residue $R$ is a geometric hyperplane when $R$ has type $\op(i)$.
\end{Proposition}

Note that $w_0$ acts like the identity on all diagrams, but flips
$A_n$, $D_{2m+1}$, $E_6$. So, we always find a geometric hyperplane,
except for non-middle nodes of $A_n$, for the dual polar geometries
for $D_{2m+1}$, for $E_{6,1}$ and $E_{6,2}$.

Now in a point-line geometry we have at least two natural concepts
of `far'. Is it true that being opposite is equivalent to being
at maximal distance in the point-line geometry?

\begin{Proposition}
Yes.
\end{Proposition}
\Proof
First of all, we may restrict ourselves to the thin case.
Look at the double coset diagram of the point-line geometry.
The distance from $1$ of the double coset represented by $w$ is the minimum
number of factors $r$ (where $r$ is the reflection belonging to node $i$)
in any expression of $w$. Since $w < w_0$ in Bruhat order, for each
reduced expression of $w_0$ there is a reduced expression for $w$
obtained from that of $w_0$ by cancelling factors.
Thus, no double coset is farther from $1$ than $w_0$.
If $w_0$ does leave type $i$ invariant, then the double coset diagram
of the point-line geometry is reflected by left multiplication by $w_0$
(see \cite{BCN}, 10.2.11). Since the double coset of $1$ has a unique
neighbour in the diagram, this means that also the double coset of $w_0$
has a unique neighbour in the diagram.
Consider a double coset represented by $w$ different from that represented by $w_0$.
There is an expression $w_0 = wu$ with $l(w_0) = l(w) + l(u)$,
and $u$ describes a walk from $w$ to $w_0$ through the double coset diagram
in which the distance to $1$ never decreases.
But it must pass through the unique neighbour of $w_0$, which is closer to $1$,
so also $w$ is closer to $1$.

On the other hand, if $w_0$ moves type $i$, then we either have a Grassmann
graph, or a dual polar graph, or $E_{6,1}$ (all distance regular), or $E_{6,2}$.
Only the latter requires further investigation, and we find that also there
there is a unique double coset at maximal distance, see the diagram below. \qed

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{\bf Problem} Is there a nicer proof, without having to look at $E_{6,2}$?

\begin{Theorem}
In a thick near polygon with quads, the complement of a hyperplane
is connected.
\end{Theorem}
\Proof
Let $H$ be a hyperplane, and $x,y$ two points outside.
Let $i = d(x,y)$. We show that the distance can be diminished
while walking in the complement of $H$.
Pick a line $L$ on $x$ that contains a point $x'$ with $d(x',y) = i-1$.
Pick a line $M$ on $y$ parallel to $L$. Then $M$ contains a point
$y'$ with $d(y',x) = i-1$. If either $x'$ or $y'$ is outside $H$,
then we are done. Otherwise, we can pick a third point $x''$
on $L$; it has distance $i-1$ to some point $y''$ of $M$,
and $y'' \ne y'$ so that $x'',y'' \notin H$.
Thus, walking from $x$ to $x''$ and from
$y$ to $y''$ we have diminished the distance. \qed

\begin{Theorem}
In a point-line geometry $\Gamma$ defined by selecting type $i$
in a spherical building, the complement of a hyperplane is connected
(and has diameter at most $2d-1$ when $\Gamma$ has diameter $d$)
when $\mu$ (the number of points collinear to each of two points
at mutual distance 2) is larger than 1.
\end{Theorem}

Thus, the conclusion of the theorem holds in the cases
$A_{n,i}$ ($n \ge i \ge 1$), $B_{n,1}$ ($n \ge 3$),
$B_{n,n}$ ($n \ge 2$), $D_{n,1}$ ($n \ge 4$),
$D_{n,n}$ ($n \ge 4$), $E_{6,1}$, $E_{7,7}$.

\Proof
Similar to the above, using the existence of `parallel' lines.
In Weyl group terms this means that
$$i^W = i^{W_{I \setminus \{i\}}} .$$
(The same condition is necessary and, in cases with single bonds only,
sufficient to have an apartment generate the geometry.) \qed


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{\it Groupes et alg\`ebres de Lie, Chap. IV, V, VI\/},
Hermann, Paris, 1968.

\bibitem{B93} A.E. Brouwer,
{\it The complement of a geometric hyperplane in a generalized polygon
is usually connected\/},
pp. 53-57 in: Finite geometry and combinatorics - Proc. Deinze 1992,
F. De Clerck et al., London Math. Soc. Lect. Note Ser. 191,
Cambridge Univ. Press, 1993.

\bibitem{BCN} A.E. Brouwer, A.M. Cohen \& A. Neumaier,
{\it Distance-regular graphs\/},
Ergebnisse der Mathematik 3.18, Springer, Heidelberg, 1989.

\bibitem{Bue81} F. Buekenhout,
{\it The basic diagram of a geometry\/},
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Lecture Notes in Math. 893, Springer, Berlin, 1981.

\bibitem{MR} B. M\"uhlherr \& M. Ronan,
{\it Local to global structure in twin buildings\/},
Invent. Math. {\bf 122} (1995) 71-81.

\bibitem{Ron89} M. Ronan,
{\it Lectures on buildings\/},
Academic Press, Boston etc., 1989.

\bibitem{Tits86a} J. Tits,
{\it Buildings of spherical type and finite BN-pairs\/},
Lecture Notes in Math. 386, Springer, Berlin, 1974.

\bibitem{Tits81} J. Tits,
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C. Davis, B. Gr\"unbaum, F.A. Sherk (eds.),
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\end{thebibliography}

\bigskip
\noindent{\bf R. J. Blok}
Department of Mathematics,
Delft University of Technology,
P.O. Box 5031, 2600 GA \ Delft, The Netherlands.\\
 e-mail: R.J.Blok@twi.tudelft.nl

\bigskip
\noindent{\bf A. E. Brouwer}
Department of Mathematics,
Eindhoven University of Technology,
P.O. Box 513, 5600 MB \ Eindhoven, The Netherlands.\\
 e-mail: aeb@cwi.nl

\end{document}


