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\begin{document}
\title{Buekenhout-Tits geometries and chain calculus}
\date{}
\maketitle

\begin{abstract}
We say some introductory words on Buekenhout-Tits geometries
and describe chain calculus in $A_n$, $D_n$, $E_6$, $E_7$, $E_8$.
\end{abstract}

\section{Buekenhout-Tits geometries}
A {\em Buekenhout-Tits geometry} is a set $X$ of {\em objects}
provided with a symmetric relation $*$ called {\em incidence}
and a function $t:X \rightarrow I$ that assigns a {\em type}
to each object, such that two objects of the same type are never incident.
The set $I$ is the set of types. The cardinality $|I|$
is called the {\em rank} of the geometry.

Let us call the geometry $\Gamma(X,*,I,t)$.

A Buekenhout-Tits geometry can be viewed as a multipartite graph $\Gamma$
with vertex set $X$ and partition $\{X_i \mid i \in I\}$
(with $X_i = t^{-1}(i)$), with incidence taken as adjacency.

(Also common is the slightly different definition that adds a loop
at every vertex of $\Gamma$, so that two objects of the same type
are incident if and only if they coincide. It will not make any difference
whether we draw a loop at every vertex or at no vertex.)

The geometry $\Gamma$ is called {\em connected} when the graph $\Gamma$
is connected. (Note that the graph without vertices is not connected:
a connected graph has precisely 1 connected component, while the graph
without vertices has no connected component.)

A {\em flag} $F$ in $\Gamma$ is a clique, a complete subgraph,
a set of mutually incident objects. No two elements of a flag
have the same type. The {\em rank} of $F$ is $|t(F)|$ (that is, $|F|$).
The {\em corank} of $F$ is $| I \setminus t(F)|$.

The {\em residue} ${\rm Res}(F)$ (also written $\Gamma_F$) is the geometry
with set of objects $Y = \{y \in X \setminus F \mid F \cup \{y\}
~\mbox{\rm is a flag}\}$, incidence inherited from $\Gamma$,
set of types $I \setminus t(F)$, and type function inherited from $\Gamma$.

The geometry $\Gamma$ is called {\em residually connected} when every residue
of rank at least two is connected (and hence nonempty), and every residue
of rank one is nonempty.

\medskip
The intuition that belongs to a Buekenhout-Tits geometry
is that of a collection of geometrical objects of various types
(points, lines, planes, circles, ...) together with some incidence
between them. An axiom system is imposed by giving a Buekenhout-Tits
diagram.

\subsection{Buekenhout-Tits diagrams}
Let ${\cal D}$ be a labelled graph on $I$, where for $i,j \in I$
the label ${\cal D}_{ij}$ is a class of rank 2 geometries.
We say that ${\cal D}$ is a Buekenhout-Tits diagram for
the geometry $\Gamma = (X,*,I,t)$
when for every flag $F$ of $\Gamma$ of corank 2,
say $t(F) = I \setminus \{ i,j\}$, the residue $\Gamma_F$
belongs to the class of geometries ${\cal D}_{ij}$.

This is a recursive definition of the meaning of a diagram
in terms of what the labelled edges mean for rank 2 geometries.

There is a dictionary of traditional labels.

\begin{picture}(35,10)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\end{picture}: Every $i$-object is incident to every $j$-object.

\begin{picture}(35,10)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\put(0,0){\line(1,0){24}}
\end{picture}: The $i$-objects and $j$-objects form the
points and lines of a projective plane.

\begin{picture}(35,10)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\put(0,1){\line(1,0){24}}
\put(0,-1){\line(1,0){24}}
\end{picture}: The $i$-objects and $j$-objects form the
points and lines of a generalized quadrangle.

\begin{picture}(35,10)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\put(0,-1.5){\line(1,0){24}}
\put(0,0){\line(1,0){24}}
\put(0,1.5){\line(1,0){24}}
\end{picture}: The $i$-objects and $j$-objects form the
points and lines of a generalized hexagon.

\begin{picture}(35,12)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\put(12,0){\makebox(0,10){\small Af}}
\put(0,0){\line(1,0){24}}
\end{picture}: The $i$-objects and $j$-objects form the
points and lines of an affine plane.

\begin{picture}(35,12)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\put(12,0){\makebox(0,10){\small C}}
\put(0,0){\line(1,0){24}}
\end{picture}: The $i$-objects and $j$-objects form the
points and edges of a complete graph.

Etc.

\medskip
{\bf Examples}
The geometry of points, lines and planes in a 3-dimensional
projective space satisfies the axioms given by the diagram
\begin{picture}(60,10)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\put(48,0){\circle*{3.5}}
\put(0,0){\line(1,0){48}}
\end{picture}.

The geometry of points, lines and planes in a 3-dimensional
affine space satisfies the axioms given by the diagram
\begin{picture}(60,12)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(12,0){\makebox(0,10){\small Af}}
\put(24,0){\circle*{3.5}}
\put(48,0){\circle*{3.5}}
\put(0,0){\line(1,0){48}}
\end{picture}.

The geometry of 8 corners, 12 edges and 6 faces of a cube
satisfies the axioms given by the diagram
\begin{picture}(60,12)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(0,1){\line(1,0){24}}
\put(0,-1){\line(1,0){24}}
\put(24,0){\circle*{3.5}}
\put(24,0){\line(1,0){24}}
\put(48,0){\circle*{3.5}}
\end{picture}.

The geometry of totally singular points, lines, planes and solids
in a geometry of type $O_8^+(F)$ satisfies the axioms given by the diagram
\begin{picture}(84,12)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(0,0){\line(1,0){48}}
\put(24,0){\circle*{3.5}}
\put(48,0){\circle*{3.5}}
\put(48,1){\line(1,0){24}}
\put(48,-1){\line(1,0){24}}
\put(72,0){\circle*{3.5}}
\end{picture}.

The geometry of totally singular points, lines,
solids of the first kind, and solids of the second kind,
in a geometry of type $O_8^+(F)$ satisfies the axioms given by the diagram
\begin{picture}(60,30)(-5,-10)
\put(0,0){\circle*{3.5}}
\put(24,0){\circle*{3.5}}
\put(41,17){\circle*{3.5}}
\put(41,-17){\circle*{3.5}}
\put(0,0){\line(1,0){24}}
\put(24,0){\line(1,1){17}}
\put(24,0){\line(1,-1){17}}
\end{picture}.



\subsection{Simple properties}

\begin{Proposition}
Let $\Gamma$ be a residually connected Buekenhout-Tits geometry
of finite rank.

(i) For any two distinct types $i,j \in I$ the subgraph
induced on $X_i \cup X_j$ is connected.

(ii) If the types $i,j$ belong to different connected components
of the Buekenhout-Tits diagram, then each $i$-object is incident
with each $j$-object. \qed
\end{Proposition}

\Proof
(i) Induction on the rank. The case of rank at most 2 holds by definition.
Since $\Gamma$ is connected, we can join two objects in $X_i \cup X_j$
by a chain $x_0 * x_1 * ... * x_l$. Next, for each $x_h$ in this chain
with a type different from $i$ and $j$, we can replace $x_h$ by
a chain in $X_i \cup X_j$ in ${\rm Res}(x_h)$ (by the induction hypothesis
and residual connectedness).

(ii) Induction on the rank. The case of rank at most 2 holds by definition.
Using part (i) we can join two objects $x \in X_i$ and $y \in X_j$
by a chain $x = x_0 * x_1 * ... * x_l = y$ contained in $X_i \cup X_j$
(so that the types alternate between $i$ and $j$). Let the length $l$
be chosen minimal, and suppose that $l > 1$. Let $k$ be a third type
different from $i$ and $j$. We may suppose that $j$ and $k$ belong to
different connected components of the Buekenhout-Tits diagram.
In ${\rm Res}(x_1)$ we can replace $x_0 * x_1 * x_2$ by a path
$x_0 = x'_0 * x'_1 * ... * x'_m = x_2$ using only types $i$ and $k$.
Now $x_3$ and its two predecessors in the chain have types
$k$-$i$-$j$, and by the induction hypothesis we can omit the
middle object (of type $i$). Then $x_3$ and its two predecessors
have types $i$-$k$-$j$, and again we can omit the middle object.
It follows after $m$ steps that $x_0 * x_3$, so that $l$ was not minimal.
\qed

\medskip
After this preparation, it is an easy exercise to prove the Veblen-Young
axiom from the $A_n$ diagram, so that a (thick) geometry satisfying
the $A_n$ diagram is a projective space.

\section{Chain calculus}

\subsection{Existence of chains}

We shall talk about chains $x_0 * x_1 * ... * x_l$
(in some residually connected Buekenhout-Tits geometry satisfying
a given diagram) by just giving the sequence of types
$t_0$-$t_1$-...-$t_l$, where the object $x_i$ is of type $t_i$.

A sequence of types given as a statement, denotes the claim that
arbitrary objects $x_0$ and $x_l$ of the types occurring
first and last can be joined by a chain of objects of the indicated
types, each incident with the preceding and following.
In the proofs we shall modify chains, but always keep the ends fixed.

\medskip
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\put(48,-6){\makebox(0,0){\small 3}}
\put(0,0){\line(1,0){48}}
\put(60,0){\makebox(0,0){\ldots}}
\put(72,0){\circle*{3.5}}
\put(72,-6){\makebox(0,0){\small $n\minns 1$}}
\put(72,0){\line(1,0){24}}
\put(96,0){\circle*{3.5}}
\put(96,-6){\makebox(0,0){\small $n$}}
\end{picture} $A_n$.


\bigskip\noindent{\bf Proposition} $A_n$:
{\em
For $2 \le i \le n$ we have $1$-$i$-$(i-1)$.
In particular, for $n \ge 2$, we have $1$-$2$-$1$.}

\Proof
If $i < n$, then by induction we find that if
$1$-$2$-$1$-$i$-$(i-1)$, then
$1$-$2$-$(i+1)$-$i$-$(i-1)$, hence
$1$-$(i+1)$-$(i-1)$, hence
$1$-$i$-$(i-1)$,
so chains $1$-$(2$-$1$-$)^j~i$-$(i-1)$ can be shortened
to $1$-$i$-$(i-1)$, and by residual connectedness we are done.
By definition of $A_2$ we have $1$-$2$-$1$ in $A_2$.
Remains the case $i = n \ge 3$. But there we have
$1$-$2$-$1$-$(n-1)$, so
$1$-$2$-$n$-$(n-1)$, so
$1$-$n$-$(n-1)$,
by induction and since $1$-$2$-$1$ holds. \qed

\medskip
\begin{picture}(110,30)(-5,-13)
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\put(48,0){\circle*{3.5}}
\put(48,-6){\makebox(0,0){\small 3}}
\put(0,0){\line(1,0){48}}
\put(60,0){\makebox(0,0){\ldots}}
\put(72,0){\circle*{3.5}}
\put(72,-6){\makebox(0,0){\small $n\minns 2$}}
\put(72,0){\line(1,1){17}}
\put(72,0){\line(1,-1){17}}
\put(89,17){\circle*{3.5}}
\put(89,-17){\circle*{3.5}}
\put(89,-23){\makebox(0,0){\small $n\minns 1$}}
\put(89,23){\makebox(0,0){\small $n$}}
\end{picture} $D_n$.


\bigskip\noindent{\bf Proposition} $D_n$:
{\em
Let $n \ge 2$. Then the following hold.

(a) $1$-$(n-1)$-$n$.

(b) $1$-$i$-$(i-1)$-$i$ for $2 \le i \le n-2$.
In particular: $1$-$2$-$1$-$2$.

(c) If $n$ is even, then $(n-1)$-$1$-$n$.
If $n$ is odd, then $n$-$1$-$n$.
}

\Proof
In $D_2$ we have $1$-$2$, implying all our claims.
For $n = 3$ everything follows from Proposition $A_3$.
Now use induction on $n$.
For part (a) we find by induction and Proposition $A_n$: if
$1$-$2$-$1$-$(n-1)$-$n$ then
$1$-$2$-$n$-$(n-1)$-$n$ so
$1$-$n$-$(n-2)$-$n$ so
$1$-$(n-1)$-$(n-2)$-$n$ so
$1$-$(n-1)$-$n$, proving part (a).

For part (b):
$1$-$(n-1)$-$n$-$i$, so
$1$-$(n-1)$-$(i-1)$-$i$, so
$1$-$i$-$(i-1)$-$i$.

For part (c): if $n$ is even, then (by induction):
$(n-1)$-$n$-$1$-$n$, so
$(n-1)$-$n$-$2$-$n$, so
$(n-1)$-$1$-$2$-$n$, so
$(n-1)$-$1$-$n$,
and if $n$ is odd, then
$n$-$1$-$(n-1)$-$n$, so
$n$-$2$-$(n-1)$-$n$, so
$n$-$2$-$1$-$n$, so
$n$-$1$-$n$. \qed

For $E_6$, $E_7$, $E_8$ we shall omit the `-' in type sequences.

\medskip
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\put(72,-6){\makebox(0,0){\small 4}}
\put(96,0){\circle*{3.5}}
\put(96,-6){\makebox(0,0){\small 5}}
\end{picture} $E_6$.

\bigskip\noindent{\bf Proposition} $E_6$:
{\em
(a) $151$,

(b) $12165$,

(c) if $1215$ then $165$.
}

\Proof
(c) $1215$ yields $1265$ and then $165$.

(a) $12151$ yields $1651$, $1641$, $1541$, $151$.

(b) $1515$ yields $14615$, $14625$, $121625$, $121325$,
$121365$, $12165$. \qed

\medskip
\begin{picture}(134,40)(-5,-3)
\put(0,0){\circle*{3.5}}
\put(0,-6){\makebox(0,0){\small 1}}
\put(0,0){\line(1,0){120}}
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\put(96,-6){\makebox(0,0){\small 5}}
\put(120,0){\circle*{3.5}}
\put(120,-6){\makebox(0,0){\small 6}}
\end{picture} $E_7$.

\bigskip\noindent{\bf Proposition} $E_7$:
{\em
(a) $1216$,

(b) $1612$,

(c) if $12121$ then $161$.
}

\Proof
(c) $12121$ yields $12621$ and then $161$.

(a) $121216$ yields $1616$, $1626$, $12126$, $1216$.

(b) $12121212$ yields $161212$, $161262$, $16162$, $16262$,
$121262$, $12162$, $121212$, $1612$. \qed

\medskip
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\put(120,-6){\makebox(0,0){\small 6}}
\put(144,0){\circle*{3.5}}
\put(144,-6){\makebox(0,0){\small 7}}
\end{picture} $E_8$.

\bigskip\noindent{\bf Proposition} $E_8$:
{\em
(a) $17171$,

(b) if $17121$ then $12121$,

(c) $121212$.
}

\Proof
(b) $17121$ yields $172321$, $17231$, $121231$, $12131$, $12121$.

(a) $1717121$ yields $1712121$, $1212121$, $121232721$,
$1213271$, $12723271$, $1721271$, $17171$.

(c) $12121212$ yields $1212123272$, $121213272$, $1212723272$,
$1217172$, (by (b)) $1212172$, $12123272$, $1213272$,
$12723272$, $1723272$, $17172$, $171212$, $121212$. \qed

For the collinearity graph $\Gamma$ (vertices: objects of type 1;
adjacency: both in the residue of some flag of cotype 1 -
in our cases this is equivalent to both incident to some object
of type 2) the above means the following:

$A_n$: $\Gamma$ is a clique (has diameter 1)

$D_n$: $\Gamma$ has diameter 2; any line carries a point at distance
at most one from a given point

$E_6$: $\Gamma$ has diameter 2 - indeed, any two vertices are in a
$D_5$ subgraph

$E_7$: $\Gamma$ has diameter 3; any two vertices at distance 2
are in a $D_6$ subgraph; any line carries a point at distance
at most two from a given point

$E_8$: $\Gamma$ has diameter 3; if $x$ and $y$ are two points
at distance 2 in a $D_7$ subgraph, then $y$ has no neighbours
at distance 3 from $x$; any line carries a point at distance at most
two from a given point

\medskip
For the relation between points $x$ and symplecta $S$
(objects of type $m-1$) in $E_m$, the above implies:

$E_6$: $x^\perp \cup S$ is either empty or a projective 4-space.

$E_7$: $x^\perp \cup S$ is either a single point or a
projective 5-space.

$E_8$: $x^\perp \cup S$ is either empty or a line or a
projective 6-space.


% \subsection{Uniqueness of chains}
% TBA.


\end{document}
