---
abstract: 'We give a uniform construction of free pseudospaces of dimension $n$ extending work in [@BP]. This yields examples of $\omega$-stable theories which are $n$-ample, but not $n+1$-ample. The prime models of these theories are buildings associated to certain right-angled Coxeter groups.'
author:
- Katrin Tent
title: 'The free pseudospace is $n$-ample, but not $(n+1)$-ample '
---

Introduction
============

In the investigation of geometries on strongly minimal sets the notion of ampleness plays an important role. Algebraically closed fields are $n$-ample for all $n$ and it is not known whether there are strongly minimal sets which are $n$-ample for all $n$ and do not interpret an infinite field. Obviously, one way of proving that no infinite field is interpretable in a theory is by showing that the theory is *not* $n$-ample for some $n$.

In [@BP], Baudisch and Pillay constructed a free pseudospace of dimension $2$. Its theory is ${\omega}$-stable (of infinite rank) and $2$-ample. F. Wagner posed the question whether this example was $3$-ample or not.

In Section \[sec:construction\] we give a uniform construction of a free pseudospace of dimension $n$ and show that it is $n$-ample, but not $n+1$-ample. It turns out that the theory of the free pseudospace of dimension $n$ is the first order theory of a Tits-building associated to a certain Coxeter diagram and we will investigate this connection in Section \[sec:building\].

In the final section we show that there are exactly two orthogonality classes of regular types.

Construction and results {#sec:construction}
========================

Fix a natural number $n\geq 1$. Let $L_n$ be the language for $n+1$-coloured graphs containing predicates $V_i,i=0,\ldots n$ and an edge relation $E$.

By an $L_n$-graph we mean an $n+1$-coloured graph with vertices of types $V_i, i=0,\ldots n$ and an edge relation $E\subseteq \bigcup_{i=1,\ldots n}V_{i-1}\times V_i$. We say that a path in this graph is of type $E_i$ if all its vertices are in $V_{i-1}\cup V_i$ and of type $E_i\cup \ldots \cup E_{i+j}$ if all its vertices are in $V_{i-1}\cup\ldots\cup V_{i+j}$

The free pseudospaces will be modeled along the lines of a projective space, i.e. we will think of vertices of type $V_i$ as $i$-dimensional spaces in a . Therefore we extend the notion of incidence as follows:

\[d:incidence\]

1.  \[d:incidence1\] We say that a vertex $x_i$ of type $V_i$ is *incident* to a vertex $x_j$ of type $V_j$ if there are vertice $x_l$ of type $V_l, l=i+1\ldots j$ such that $E(x_{l-1},x_l)$ holds. In this case the sequence $(x_i,\ldots x_j)$ is called a *dense flag*. A *flag* is a sequence of vertices $(x_1,\ldots x_k)$ in which any two vertices are incident.

2.  The *residue* $R(x)$ of a vertex $x$ is the set of vertices incident with $x$.

3.  We say that two vertices $x$ and $y$ *intersect in the vertex* $z$ and write $z=x\wedge y$ if the set of vertices of type $V_0$ incident with $x$ and $y$ is exactly the set of vertices of type $V_0$ incident with $z$. If there is no vertex of type $V_0$ incident to $x$ and $y$, we say that $x$ and $y$ intersect in the empty set.

4.  We say that two vertices $x$ and $y$ *generate the vertex* $z$ and write $z=x\vee y$, if the set of vertices of type $V_n$ incident with $x$ and $y$ is exactly the set of vertices of type $V_n$ incident with $z$. If there is no vertex of type $V_n$ incident to $x$ and $y$, we say that $x$ and $y$ generate the empty set.

5.  A *simple cycle* is a cycle without repetitions.

We now give an inductive definition of a of dimension $n$:

A of dimension $1$ is a free pseudoplane, i.e. an $L_1$-graph which does not contain any cycles and such that any vertex has infinitely many neighbours.

Assume that a free pseudospace of dimension $n-1$ has been defined. Then a of dimension $n$ is an $L_n$-graph such that the following holds:

1.  1.  The set of vertices of type $V_0\cup \ldots \cup V_{n-1}$ is a .

    2.  The set of vertices of type $V_1\cup \ldots \cup V_n$ is a free pseudospace of dimension $(n-1)$.

2.  1.  For any vertex $x$ of type $V_0$, $R(x)$ is a free pseudospace of dimension $(n-1)$.

    2.  For any vertex $x$ of type $V_n$, $R(x)$ is a free pseudospace of dimension $(n-1)$.

3.  1.  Any two vertices $x$ and $y$ intersect in some vertex $z$ or the emptyset.

    2.  Any two vertices $x$ and $y$ generate some vertex $z$ or the emptyset.

4.  1.  If $a$ is a vertex of type $V_n$ and $\gamma=(a,b,\ldots , b',a)$ is a simple cycle of length $k$, then there is an $E_1\cup\ldots \cup E_{n-1}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ all of whose $V_0$-vertices appear in $\gamma$.

    2.  If $a$ is a vertex of type $V_0$ and $\gamma=(a,b,\ldots , b',a)$ is a simple cycle of length $k$, then there is an $E_2\cup\ldots \cup E_n$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ all of whose $V_n$-vertices appear in $\gamma$.

Let $T_n$ denote the $L_n$-theory expressing these axioms.

Note that the inductive nature of the definition immediately has the following consequences:

1.  The induced subgraph on $V_j\cup\ldots\cup V_{j+m}$ is a free pseudospace of dimension $m$.

2.  If $a$ is a vertex of type $V_{j+m}$ and $\gamma=(a,b,\ldots , b',a)$ is a simple cycle of length $k$ contained in $V_j\cup\ldots\cup V_{j+m}$, then there is an $E_{j+1}\cup\ldots \cup E_{j+m-2}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ all of whose $V_j$-vertices appear in $\gamma$.

3.  The notion of a is *self-dual*: if we put $W_i=V_{n-i}, i=0,\ldots n$, then $W_0,\ldots W_n$ with the same set of edges is again a .

Our first goal is to show that $T_n$ is consistent and complete.

Let $A$ be a finite $L_n$-graph. The following extensions are called elementary strong extensions of $A$:

1.  add a vertex of any type to $A$ which is connected to at most one vertex of $A$ of an appropriate type.

2.  If $(x,y,z)$ is a dense flag in $A$, add a vertex of the same type as $y$ to $A$ which is connected to both $x$ and $z$.

We write $A\leq B$ if $B$ arises from $A$ by finitely many elementary strong extensions.

Let $\K_n$ be the class of finite $L_n$-graphs $A$ such that the following holds

1.  $A$ does not contain any $E_i$-cycles for $i=1,\ldots n$.

2.  If $a\neq a'$ are in $A$, they intersect in a vertex of $A$ or the emptyset.

3.  If $a\neq a'$ are in $A$, they generate a vertex of $A$ or the emptyset.

4.  If $(b,a,b')$ is a path with $a\in V_i,b,b'\in V_{i-1}$, and $\gamma=(a,b,\ldots , b',a)$ is an $E_i\cup E_{i-j}$ path of length $k$, then there is some $E_{i-1}\cup \ldots E_{i-j}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ with all $V_{i-j}$-vertices occurring in $\gamma$.

5.  If $(b,a,b')$ is a path with $a\in V_i,b,b'\in V_{i+1}$, and $\gamma=(a,b,\ldots , b',a)$ is an $E_i\cup\ldots\cup E_{i+j}$ path of length $k$, then there is some $E_{i+1}\cup\ldots \cup E_{i+j}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ with all $V_{i+j}$-vertices occurring in $\gamma$.

Note that $\K_n$ is closed under finite substructures. We next show that $(\K,\leq)$ has the amalgamation property for strong extensions.

For any finite $L_n$-graphs $A\subseteq B,C$ we denote by $B\otimes_AC$ the *free amalgam* of $B$ and $C$ over $A$, i.e. the graph on $B\cup C$ containing no edges between elements of $B\setminus A$ and $C\setminus A$.

If $A\leq B,C$ are in $\K_n$, then $D:=B\otimes_AC\in\K$ and $B,C\leq D$.

Clearly, $B,C\leq D$. To see that $D\in\K_n$, note that if $B\in \K_n$ and $B'$ is an elementary strong extension of $B$, then also $B'\in\K_n$. This is clear for strong extensions of type 1. For strong extensions of type 2. suppose that $(b,a,b')$ is a path with $a\in V_i,b,b'\in V_{i-1}$, and $\gamma=(a,b,\ldots , b',a)\subset B'$ is an $E_i\cup\ldots\cup E_{i-j}$-path of length $k$ containing the new vertex $y$. Since the new vertex has exactly two neighbours $y_1,y_2$, this implies that the vertex is of type $V_m$ for some $i-j\leq m\leq i$ and $(y_1,y,y_2)$ is contained in $\gamma$. By construction of strong extensions, there is some $z\in B$ such that $(y_1,z,y_2)$ is a path. Hence we may replace all occurrences of $y$ in $\gamma$ by $z$. Then $\gamma$ is contained in $B$ and we find the required path in $R(a)$ with all $V_{i-j}$-vertices occurring in $\gamma$.

This shows that the class $(\K_n,\leq)$ has a Hrushovski limit $M_n$, i.e. a countable $L_n$-structure $M_n$ whose strong subsets are exactly the $L_n$-graphs in $\K_n$ and which is homogeneous for strong subsets: if $A,B\leq M_n$ then any isomorphism from $A$ to $B$ extends to an automorphism of $M_n$. Here we say as usual that a subset $A$ of $M_n$ is strong in $M_n$ if $A\cap B\leq B$ for any finite set $B\subset M_n$.

\[p:pseudospace\] The Hrushovski limit $M_n$ is a model of $T_n$.

By construction, $V_i\cup\ldots\cup V_{i+j}$ satisfies $(\Sigma 3)_j$ and $(\Sigma 4)_j$ for any $i,j$. In particular, $M_n$ satisfies $(\Sigma 3)_n$ and $(\Sigma 4)_n$.

[**$(\Sigma 1)_n$:**]{} In order to show that $M$ satisfies $(\Sigma 1)_n$, we first note that $V_i\cup V_{i+1}$ is a free pseudoplane for all $i=0,\ldots n-1$. Assume inductively that $V_j\cup\ldots \cup V_{j+i}$ is a free pseudospace of dimension $i$. To see that $V_j\cup\ldots \cup V_{j+i+1}$ is a free pseudospace of dimension $i+1$, we need only verify $(\Sigma 2)_{i+1}$. Hence we have to show that for $a\in V_j$ the residue $R(a)\cap (V_j\cup\ldots \cup V_{j+i+1})$ is a free pseudospace of dimension $i$. We know by induction that $R(a)\cap (V_j\cup\ldots \cup V_{j+i})$ is a free pseudospace.

Clearly, $$R(a)\cap (V_{j+1}\cup\ldots \cup V_{j+i+1})=\bigcup \{R(b)\cap (V_{j+1}\cup\ldots \cup V_{j+i+1})\colon b\in V_{j+1}, E(a,b) \}.$$ For each neighbour $b\in V_{j+1}$ of $a$, the set $R(b)\cap (V_{j+1}\cup\ldots \cup V_{j+i+1})$ is a free pseudospaces of dimension $i-1$ by induction. Since $(V_{j+1}\cup\ldots \cup V_{j+i+1})$ is a free pseudospace of dimension $i$, $(\Sigma 2)_{i+1}$ follows from the induction hypothesis. Hence $V_0\cup\ldots \cup V_{n-1}$ and $V_1\cup\ldots \cup V_n$ are free pseudospaces of dimension $n-1$.

[**$(\Sigma 2)_n$:**]{} The proof of $(\Sigma 2)_n$ is similar.

We say that a model $M$ of $T_n$ is $\K_n$-saturated if for all finite $A\leq M$ and strong extensions $C$ of $A$ with $C\in\K_n$ there is a strong embedding of $C$ into $M$ fixing $A$ elementwise. Clearly, by construction, $M_n$ is $\K_n$-saturated.

\[l:K-sat\] A model $M$ of $T_n$ is $\omega$-saturated if and only if $M$ is $\K_n$-saturated.

Let $M$ be an $\omega$-saturated model of $T_n$. To show that $M$ is $\K_n$-saturated, let $A\leq M$ and $A\leq B\in\K_n$. By induction we may assume that $F$ is an elementary strong extension of $A$ and it is easy to see that $F$ can be imbedded over $A$ into $M$.

The theory $T_n$ is complete.

Let $M$ be a model of $ T_n$. In order to show that $M$ is elementarily equivalent to $M_n$ choose an $\omega$-saturated $M'\equiv M$. By Lemma \[l:K-sat\], $M'$ is $\K_n$-saturated. Now $M'$ and $M_n$ are partially isomorphic and therefore elementarily equivalent.

We will see in Section \[sec:building\] that $T_n$ is the theory of the building of type $A_{\infty, n+1}$ with infinite valencies.

Following *[@BP]* we call a subset $A$ of a model $M$ of $T_n$ *nice* if

1.  any $E_i$-path between elements of $A$ lies entirely in $A$ and

2.  if $a,b\in A$ are connected by a path in $M$ there is a path from $a$ to $b$ inside $A$.

\[l:strong=nice\]\[r:qftp\] Note that a subset $A$ of $M_n$ is strong in $M_n$ if and only if it is nice. (This follows immediately from the definition of strong extension.)

Since $M_n$ is homogeneous for strong subsets, for any nice subset of $M_n$ the quantifier-free type determines the type, i.e. if $\bar a,\bar b$ are nice in $M_n$ and such that $\operatorname{qftp}(\bar a)=\operatorname{qftp}(\bar b)$, then $\operatorname{tp}(\bar a)=\operatorname{tp}(\bar b)$.

We now work in a very saturated model $\overline{M}$ of $T_n$.

\[l:strongext\] If $A$ is a finite set, there is a nice finite set $B$ containing $A$.

Since single vertices are nice it suffices to prove the following

[**Claim:**]{} *If $A$ is nice and $a$ arbitrary, then there is a nice finite set $B$ containing $A\cup\{a\}$.*

*Proof of Claim:* Of course we may assume $a\notin A$. If there is no path from $a$ to $A$, clearly $A\cup\{a\}$ is nice. Hence we may also assume that there is some path $\gamma=(a=x_0,\ldots b)$ for some $b\in A$ and $\gamma\cap A=\{b\}$. It therefore suffices to prove the claim for the case where $a$ has a neighbour in $A$. If $a$ has two neighbours $x,y\in A$ then $(x,a,y)$ is a dense flag and $A\cup\{a\}$ is nice.

Now assume that $a\in V_i$ has a unique neighbour of type $V_{i+1}$ in $A$. (The other case then follows by self-duality.) If the $E_i$-connected component of $a$ does not intersect $A$, then again $A\cup\{a\}$ is nice. Otherwise there is some $E_i$-path $\gamma=(x_0=a,\ldots x_m=b)$ in $M_n$ with $\gamma\cap A=\{b\}$. If for some $V_{i-1}$-vertex $x_k$ of $\gamma$ there is an $E_{i-1}$-path to some $c\in A$, then the $E_i$-path from $c$ to $b$ extends $(x_k,\ldots x_m=b)$ and is entirely contained in $A$ since $A$ is nice. Since $\gamma\cap A=\{b\}$, no such $x_k$ exists implying that $A\cup\gamma$ is nice.

Let us say that $\gamma$ *changes direction in $x_i$* if $x_i\in V_j$ and either $x_{i-1},x_{i+1}\in V_{j-1}$ or $x_{i-1},x_{i+1}\in V_{j-1}$ for some $j$. Clearly a path which doesn’t change direction is a dense flag.

We will use the following easy consequence of $(\Sigma 4)_{j\leq n}$.

\[l:cycle\] If $\gamma=(a,b,\ldots, b',a)$ is a simple cycle, then any two vertices $x,y\in \gamma\cap (V_{j-1}\cup V_j)$ are joined by an $E_j$-path.

Suppose that $j,j+m$ are minimal and maximal, respectively, with $\gamma\cap V_j, V_{j+m}\neq\emptyset$. Apply $(\Sigma 4)_m$ to all vertices of $\gamma\cap V_{j+m}$ and replace any path of the form $(x,y,z)$ with $y\in V_{j+m}$ by a path from $x$ to $z$ contained in $V_j\cup\ldots \cup V_{j+m-1}$. This yields a new path contained in $V_j\cup\ldots \cup V_{j+m-1}$. Next apply $(\Sigma 4)_{m-1}$ to each vertex of type $V_j$ to obtain a path in $V_{j+1}\cup\ldots \cup V_{j+m-1}$. Alternating between replacing the top and bottom extremal peaks by a path using fewer levels, we end up with an $E_k$-path for some $j\leq k\leq j+m-1$ between vertices of $\gamma$. Since an $E_k$-path changes direction at every vertex, we can apply the same procedure in order to replace this path by an $E_{k'}$-path for any $j\leq k'\leq j+m-1$.

\[def:reduced\] We call a path $\gamma=(x_0,\ldots x_k)\subseteq V_j\cup\ldots \cup V_{j+m}$ *reduced* if $\gamma$ is a flag or if the following holds:

1.  if $m=1$ the path $\gamma$ is reduced if it does not contain any repetition.

2.  for any simple cycle $\gamma'$ containing $(x_i,\ldots, x_{i+q})$ replacing all $(y,x,z)$ with $x\in (x_{i+1},\ldots, x_{i+q-1})\cap V_{j+m}$ by an $E_{j+m-1}$-path in $R(x)$ yields a reduced path $(x_i,\ldots, y,\ldots z,\ldots, x_{i+q})\subseteq V_j\cup\ldots \cup V_{j+m-1}$

Note that the definition and Lemma \[l:cycle\] immediately imply the following:

\[r:reducedpath\] Suppose that every reduced path from $a$ to $b$ contains $x$ and let $\gamma_1,\gamma_2$ be paths from $a$ to $x$ and from $x$ to $b$ respectively. Then the path $\gamma_1\gamma_2$ is reduced if and only if $\gamma_1$ and $\gamma_2$ are.

If $\gamma=(a,\ldots, b)$ is a reduced path and changes direction in $x\in V_k$, then either every reduced path from $a$ to $b$ contains $x$ or the path $(y,x,z)\subseteq \gamma$ can be replaced by an $E_{k-1}$-path (or by an $E_k$-path) $(y,\ldots z)$ such that $(a,\ldots, y,\ldots, z,\ldots b)$ is still reduced.

\[l:nicereplacement\] Suppose $\gamma=(a=x_0,\ldots, x_s=b)$ is a reduced path from $a$ to $b$ which can be completed into a simple cycle $(a,x_1,\ldots, b,\ldots,a)$. If $\gamma$ changes direction in $x\in V_k$, then there is a reduced path $\gamma'=(a=y_0,\ldots, y_t=b)$ not containing $x$ such that $x$ is on the $E_{k}$-(or $E_{k+1}$-) path between appropriate elements of $\gamma'$.

To fix notation let us assume that the neighbours of $x$ in $\gamma$ are $y,z\in V_{k-1}$. By Lemma \[l:cycle\] we may replace the path $(y,x,z)$ by an $E_k$-path. This path is obviously still reduced and $x$ is on the $E_{k+1}$-path from $y$ to $z$.

\[l:nicepath\] Suppose $D$ is a nice set containing an $E_j$-path from $a$ to $b$. Suppose furthermore that $\gamma=(a=y_0,\ldots, y_t=b)$ is a reduced path such that the composition with the $E_j$-path is a simple cycle. If $\gamma$ changes direction in $c$, then $c\in D$.

Using the fact that $M_n$ is $\omega$-saturated and homogeneous for strong substructures we can now describe the algebraic closure:

\[l:acl2\] The algebraic closure $\operatorname{acl}(a,b)$ is the intersection of all nice sets containing $a,b$.

Since there are finite nice sets containing $a,b$, the intersection of all nice sets containing $a,b$ is certainly finite and invariant over $a,b$, hence contained in $\operatorname{acl}(a,b)$.

Conversely let $D$ be a nice set containing $\{a,b\}$ and $c\notin D$. If there is no reduced path between $a$ and $b$ changing direction in $c$, then $c$ has infinitely many conjugates over $ab$, so $c\notin\operatorname{acl}(ab)$.

So suppose there is a reduced path $\gamma'$ from $a$ to $b$ changing direction in $c$. Let $\gamma\subset D$ be a path from $a$ to $b$. Composing these paths we obtain a simple cycle $\gamma"$ through $c$ consisting of subpaths $\gamma_1'\subseteq \gamma_1$ of $\gamma'\subseteq\gamma$, respectively, from some $a'$ to $b'$. In particular $\gamma_1$ is not a flag. We may assume inductively that $\gamma_1'\cap D=\{a',b'\}$. If $a'\in V_j$, then possibly after exchanging the roles of $a',b'$, there is some $x\in\gamma\cap (V_{j-1}\cup V_j)$ closest to $b'$. Then $a',x$ are connected by an $E_j$-path entirely contained in $D$ and $(x,\ldots b')$ is a flag. Since the $E_j$-path changes direction in every vertex, we also find an $E_{j-1}$-path inside $D$ to the next vertex of the flag $(x,\ldots b')$. Since $\gamma_1'$ doesn’t meet $D$ except in $a',b'$, we see that ${\gamma'_1}^\smallfrown (b',\ldots, x)$ is a reduced path. Now Lemma \[l:nicepath\] implies $c\in D$, a contradiction. Hence $c\notin\operatorname{acl}(ab)$.

Note that the proof shows in fact the following:

\[r:flagcl\]A vertex $x\neq a,b$ is in $\operatorname{acl}(ab)$ if and only if there is a reduced path from $a$ to $b$ that changes direction in $x$. Hence $\operatorname{acl}(ab)=\{a,b\}$ if and only if $a,b$ is a flag or $a$ and $b$ are not connected. In fact, we have $\operatorname{dcl}(ab)=\operatorname{acl}(ab)$.

\[l:treeacl\] If $g\in\operatorname{acl}(A)$, there exist $a,b\in A$ with $x\in\operatorname{acl}(ab)$.

We may assume that $A$ is finite. By induction it suffices to prove that if $d\in\operatorname{acl}(bc),g\in\operatorname{acl}(ad)$, then $g\in\operatorname{acl}(ab)\cup\operatorname{acl}(bc)\cup\operatorname{acl}(ac)$.

By Corollary \[r:flagcl\] there is a reduced path $\gamma_1=(b,\ldots,d,\ldots, c)$ changing direction in $d$ and a reduced path $\gamma_2=(d,\ldots ,g,\ldots, a)$ changing direction in $g$. If $\gamma_1\cup\gamma_2\in V_i\cup V_{i+1}$ for some $i$, then clearly either $(a,\ldots,g\ldots,d,\ldots,b)$ or $(a,\ldots,g\ldots,d,\ldots,c)$ is reduced.

Now assume that $\gamma_1\cup\gamma_2\in V_i\cup\ldots\cup V_{i+k}$. Clearly we may assume that $d$ is not contained in every reduced path from $a$ to $b$ or in every reduced path from $a$ to $c$. Furthermore, we may reduce to the case where no vertex of $\gamma_1$ is contained in every reduced path from $b$ to $c$ and no vertex of $\gamma_2$ is contained in every reduced path from $a$ to $d$.

By Remark \[r:reducedpath\] we may therefore replace any element $x\in V_{i+k}$ in the interior of $\gamma_1$ and $\gamma_2$ by an $E_{i+k-1}$ path between its neighbours. If $d\in V_{i+k}$, we may replace $d$ as follows: the neighbour $w$ of $d$ in $\gamma_2$ either appears in the $E_{i+k-1}$ path between the neighbours of $d$ in $\gamma_1$ or that path extends to $w$ on one of its ends. We may thus replace the paths $(b,\ldots,d,\ldots, c)$ and $(a,\ldots g,\ldots, d)$ by reduced paths $(b,b_1,\ldots,w, \ldots,c_1,c)$ and $(a,a_1,\ldots w)$ contained in $V_i\cup\ldots\cup V_{i+k-1}$, except possibly for the endpoints. We may assume that at most $a\in V_{i+k}$ since we may exchange the roles of $V_i$ and $V_{i+k}$ and replace the elements of $V_i$ instead.

By induction assumption, at least one of $(a_1,\ldots w,\ldots b_1)$ and $(a_1,\ldots w,\ldots c_1)$ is reduced, and replacing the new $E_{i+k-1}$-paths by the old $E_{i+k}$-paths, this path remains reduced and changes direction in $g$. If $a,b\in V_{i+k}$, then we may exchange the roles of $V_i$ and $V_{i+k}$ and replace the elements of $V_i$ instead. Thus we see that the path remains reduced when adding the endpoint.

\[r:acl\] For any set $A$, the algebraic closure $\operatorname{acl}(A)$ is the intersection of all nice sets containing it.

Clearly, we may restrict ourselves to finite sets $A$. As before we see that the intersection of all nice sets containing $A$ is contained in $\operatorname{acl}(A)$.

For the converse, assume that $c$ is not in the intersection of all nice sets containing $A$. If $c$ was in $\operatorname{acl}(A)$, then by Lemma \[l:treeacl\] there are $a,b\in A$ with $c\in\operatorname{acl}(ab)$. Then by Lemma \[l:acl2\] $c\in\bigcap_{a,b\in D}D\subset\bigcap_{A\subseteq D}D$, a contradiction.

For any vertex $a$ and set $A$, there is a flag $C\in\operatorname{acl}(A)$ such that for any $b\in\operatorname{acl}(A)$ there is a reduced path from $a$ to $b$ passing through one of the elements of $C$.

The flag $C$ is called the *projection* from $a$ to $A$ and we write $C=\operatorname{proj}(a/A)$. Note that $\operatorname{proj}(a/A)=\emptyset$ if and only if $a$ is not connected to any vertex of $\operatorname{acl}(A)$.

Let $b_1,b_2\in\operatorname{acl}(A)$ and let $\gamma_1=(x_0=a,\ldots ,x_m=b_1)$ and $\gamma_2=(y_0=a,\ldots ,y_l=b_2)$ be reduced paths such that $i,j$ are minimal possible with $x_i,y_j\in\operatorname{acl}(A)$. By composing the initial segments of $\gamma_1$ and $\gamma_2$ and reducing we obtain a reduced path from $x_i$ to $y_j$ intersecting $\operatorname{acl}(A)$ only in $x_i,y_j$ since $x_i, y_j$ were chosen at minimal distance from $a$. By Corollary \[r:flagcl\] $x_i,y_j$ is a flag. Thus the set of such vertices forms a flag $C$.

It is now easy to show the following:

\[T:stable\] The theory $T_n$ is $\omega$-stable.

Let $M$ be a countable model and let $\bar d$ be a tuple from $\overline{M}$. Let $C\in M$ be the finite set of projections from $\bar d$ to $M$. Then the type $\operatorname{tp}(\bar d/M)$ is determined by $\operatorname{tp}(\bar d/C)$. By Lemmas \[l:strong=nice\] and \[l:strongext\], $\bar d\cup C$ is contained in a finite strong subset of $M_n$ and for such subsets the quantifier-free type determines the type by Remark \[r:qftp\]. Hence there are only countably many types over a countable model.

In fact, it is easy to see directly without counting types that $T_n$ is superstable (see Remark \[r:superstable\]).

The has weak elimination of imaginaries.

Let $a$ be a vertex and $A$ any set. Then we can choose $\operatorname{Cb}(stp(a/A))$ as the projection of $a$ on $A$. This is a finite set.

The following immediate corollary will be very useful:

\[c:forking\] The vertex $a$ is independent from $A$ over $C$ if $\operatorname{proj}(a/AC)\subseteq\operatorname{acl}(C)$. In particular, $a$ is independent from $A$ over $\emptyset$ if and only if $a$ is not connected to any vertex of $\operatorname{acl}(A)$.

\[r:superstable\] As in [@TZU] we could have defined a notion of independence on models of $T_n$ by saying $${
 A\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{C}B
}$$ if and only if $\operatorname{proj}(a/BC)\subseteq\operatorname{acl}(C)$ for all $a\in\operatorname{acl}(A)$. It is easy to see that this notion of independece satisfies the characterizing properties of forking in stable theories (see [@TZ] Ch. 8) and hence agrees with the usual one. Note that the existence of nonforking extensions follows from the construction of $M_n$ as a Hrushovski limit. Since we have just seen that for any type $\operatorname{tp}(a/A)$ there is a finite set $A_0$ such that ${
 a\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{A_0}A
}$ this shows directly (without counting types) that $T_n$ is superstable.

Using this description of forking it is easy to give a list of regular types such that any nonalgebraic type is non-orthogonal to one of these. This is entirely similar to the list given in [@BP] and we omit the details but will return to this point in Section \[sec:types\]. It is also clear from this description of forking that the geometry on these types is trivial.

Ampleness
=========

We now recall the definition of a theory being $n$-ample:

A theory $T$ eliminating imaginaries is called $n$-ample if possibly after naming parameters there are tuples $a_0,\ldots a_n$ in $M$ such that the following holds:

1.  for $i=0,\ldots n-1$ we have $$\operatorname{acl}(a_0,.\ldots a_{i-1},a_i)\cap \operatorname{acl}(a_0,.\ldots a_{i-1},a_{i+1})=\operatorname{acl}(a_0,.\ldots a_{i-1});$$

2.  ${
     a_n\mathop{\mathpalette{
     \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\mathchardef
     \nn=12854\hss$\nn$\kern1.4\wd0\hss}\hbox to 0pt{\hss$\mid$\hss}\lower.9\ht0
     \hbox to 0pt{\hss$\smile$\hss}\kern\wd0
    }{}}_{}a_0
    },$ and

3.  ${
     a_n\mathop{\mathpalette{
     \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
     \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
    }{}}_{a_i}a_0\ldots a_i
    }$  for $i=0,\ldots n-1$.

\[T:n-ample\] The theory $T_n$ is $n$-ample and any maximal flag $(x_0,\ldots x_n)$ in $M_n$ is a witness for this.

This follows immediately from the description of $\operatorname{acl}$ in Lemma \[r:acl\] and of forking in Corollary \[c:forking\].

\[T:notample\] The is not $n+1$-ample.

Suppose towards a contradiction that $a_0,\ldots a_{n+1}$ are witnesses for $T_n$ being $n+1$-ample over some set of parameters $A$. We have $${
 a_{n+1}\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\mathchardef
 \nn=12854\hss$\nn$\kern1.4\wd0\hss}\hbox to 0pt{\hss$\mid$\hss}\lower.9\ht0
 \hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{A}a_0
},$$ $${
 a_{n+1}\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{Aa_i}a_0\ldots a_i
},i=0,\ldots n.$$

By the first condition there are vertices in $\operatorname{acl}(a_0)$ and in $ \operatorname{acl}(a_{n+1})$ which are in the same connected component. Put $f_0=\operatorname{proj}(a_{n+1}/a_0A)\in\operatorname{acl}(a_0A)$ and $f_{n+1}=\operatorname{proj}(f_0/a_{n+1}A)\in\operatorname{acl}(a_{n+1}A)$.

Since $${
 a_{n+1}\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{Aa_i}a_0\ldots a_i
}, i=1,\ldots n$$ using Corollary \[c:forking\] we inductively find flags $$f_i=\operatorname{proj}(f_{n+1}/f_0f_1\ldots f_{i-1}a_iA)=\operatorname{proj}(f_{n+1}/a_iA)\in\operatorname{acl}(a_iA),i=1,\ldots n$$ such that $${
 f_{n+1}\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{f_i}f_0f_1\ldots f_i
}.$$

For $i=1,\ldots n$ we clearly have $$\operatorname{acl}(f_0,f_1.\ldots f_{i-1},f_i)\cap \operatorname{acl}(f_0,.\ldots f_{i-1},f_{i+1})\subseteq \operatorname{acl}(a_0,.\ldots a_{i-1}A).$$

By construction, there is a reduced path $\gamma=(f_0,x_1,\ldots x_k=f_{n+1})$ containing a vertex of each of the $f_i$ in ascending order. Since we cannot have a flag containing more than $n$ elements, there must be some vertex $x$ in $\gamma$ where $\gamma$ changes direction. For some $i$ we then have $x\in f_{i+1}$ or $x$ occurs in $\gamma$ between an element of $f_i$ and an element of $f_{i+1}$. By Corollary \[r:flagcl\] we have $x\in\operatorname{acl}(f_if_{i+1})\cap\operatorname{acl}(f_if_{i+2})$. Then $${
 x\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{f_i}a_0a_1,\ldots a_iA
},$$ so $x\notin\operatorname{acl}(a_0a_1,\ldots a_iA)$, a contradiction.

The proof shows that in fact the following stronger ampleness result holds:

If $a_0,\ldots a_n$ are witnesses for $T_n$ being $n$-ample, then there are vertices $b_i\in\operatorname{acl}(a_i)$ such that $(b_0,\ldots b_n)$ is a flag.

Buildings and the prime model of $T_n$ {#sec:building}
======================================

We now turn towards constructing the prime model $M_n^0$ of $T_n$ as a Hrushovski-limit. We will show that $M_n^0$ is the building associated to a right-angled Coxeter group.

For this purpose we now consider an expansion $L_n'$ of the language $L_n$ by binary function symbols $f^i_k$. For an $L_n$-graph $A$ we put $f^i_k(x,y)=z$ if $z$ is the $k^{th}$ element on a unique shortest $E_i$-path of length at least $k$ from $x$ to $y$ and $z=x$ otherwise.

We say that an $L_n$-graph $A$ is $E_i$-connected if the set $V_{i-1}(A)\cup V_i(A)$ is connected.

Let $\K'$ be the class of finite $L_n'$-graphs $A\in\K$ which are $E_i$-connected for $i=1,\ldots n$ and additionally satisfy the following condition:

-   If $a\in A$ is of type $V_j$, then the residue $R(a)$ is $E_i$-connected for $i=1,\ldots n$.

Note that $\K'$ is closed under finitely generated substructures by the choice of language.

Let $A$ be a finite $L_n'$-graph which is $E_i$-connected for $i=1,\ldots n$. The following extensions are called elementary strong extensions of $A$:

1.  add a vertex of type $P$ or $E$ to $A$ which is connected to at most one vertex of $A$ and such that the extension is still $E_i$-connected for all $i=1,\ldots n$.

2.  If $(x,y,z)$ is a dense flag in $A$, add a vertex $y'$ of the same type as $z$ to $A$ such that $(x,y',z)$ is a flag.

3.  if $|A|\leq 1$ contains no line, add a vertex of appropriate type which is connected to the vertex of $A$ if $A\neq\emptyset$.

Again we write $A\leq B$ if $B$ arises from $A$ by finitely many elementary strong extensions.

We next show that $(\K_n',\leq)$ has the amalgamation property for $\leq$-extensions.

If $A$ contains a flag of type $(V_1,\ldots V_{n-1})$ and $A\leq B,C$ are in $\K_n'$, then $D:=B\otimes_AC\in\K_n'$ and $B,C\leq D$.

Clearly, $B,C\leq D$ and $D$ is $E_i$-connected for all $i=1,\ldots n$ since $A$ contains a flag. To see that $D\in\K$, note that if $B\in \K$ and $B'$ is an elementary strong extension of $B$, then also $B'\in\K$.

This shows that the class $(\K,\leq)$ has a Hrushovski limit $M^0_n$. Clearly, $M^0_n$ is $E_i$-connected for $i=1,\ldots n$ and since any two vertices of $M_n^0$ lie in a maximal flag, it follows that $M_n^0$ is in fact $n$-connected. Note that an $L_n'$-substructure of $M_n^0$ is automatically nice, see Remark \[r:qftp\].

The same proof as in the case of $M_n$ shows the first part of the following proposition:

\[p:primmodell\] The Hrushovski limit $M^0_n$ is a model of $T_n$ and $M^0_n$ is the unique countable model of $T_n$ which is $E_i$-connected for $i=1,\ldots n$ and such that every vertex is contained in a maximal flag.

(Note that in [@BP] the corresponding Remark 3.6 of uses Lemma 3.2, which is not correct as phrased there: $M_n^0$ and $M_n^0\cup\{a\}$ with $a$ an isolated point are not isomorphic, but satisfy the assumptions of Remark 3.6.)

The uniqueness part of Proposition \[p:primmodell\] follows directly from the following theorem and Proposition 5.1 of [@HP] which states that this type of building is uniquely determined by its associated Coxeter group and the cardinality of the residues.

\[t:building\] $M^0_n$ is a building of type $A_{\infty,n+1}$ all of whose residues have cardinality $\aleph_0$.

Recall the following definitions (see e.g. [@G]).

Let $W$ be the Coxeter group

$$W=\langle t_0,\ldots t_n\colon t_i^2=(t_it_k)^2=1, i,k=0\ldots n, |k-i|\geq 2 \rangle,$$ whose associated diagram we call $A_{\infty,n+1}$.

\[d:building\] A building of type $A_{\infty,n+1}$ is a set $\Delta$ with a *Weyl distance function* $\delta: \Delta^2\to W$ such that the following two axioms hold:

1.  For each $s\in S:=\{t_i,i=0,\ldots n\}$, the relation $x\sim_s y$ defined by $\delta(x,y)\in\{1,s\}$ is an equivalence relation on $\Delta$ and each equivalence class of $\sim_s$ has at least $2$ elements.

2.  Let $w=r_1r_2\ldots r_k$ be a shortest representation of $w\in W$ with $r_i\in S$ and let $x,y\in\Delta$. Then $\delta(x,y)=w$ if and only if there exists a sequence of elements $x,x_0,x_1,\ldots, x_k=y$ in $\Delta$ with $x_{i-1}\neq x_i$ and $\delta(x_{i-1},x_i)=r_i$ for $i=1,\ldots, k$.

A sequence as in 2. is called a *gallery of type* $(r_1,r_2,\ldots, r_k)$. The gallery is called reduced if the word $w=r_1r_2,\ldots, r_k$ is reduced, i.e. a shortest representation of $w$.

We now show how to consider $M_n^0$ as a building of type $A_{\infty,n+1}$.

(of Theorem \[t:building\]) We extend the set of edges of the $n+1$-coloured graph $M_n^0$ by putting edges between any two vertices that are incident in the sense of Definition \[d:incidence\]\[d:incidence1\]. In this way, flags of $M_n^0$ correspond to a complete subgraph of this extended graph, which thus forms a simplicial complex. A maximal simplex consists of $n+1$ vertices each of a different type $V_i$. (Such a simplex is called a *chamber*.) Let $\Delta$ be the set of maximal simplices in this graph. Define $\delta:\Delta^2\to W$ as follows:

Put $\delta(x,y)=t_i$ if and only if the flags $x$ and $y$ differ exactly in the vertex of type $V_i$. Extend this by putting $\delta(x,y)=w$ for a reduced word $w=r_1r_2\ldots r_k$ if and only if there exists a sequence of elements $x=x_0,x_1,\ldots, x_k=y$ in $\Delta$ with $x_{i-1}\neq x_i$ and $\delta(x_{i-1},x_i)=r_i$ for $i=1,\ldots, k$.

Clearly, with this definition of $\delta$, the set $\Delta$ satisfies the first condition of Definition \[d:building\]. In fact, for all $s\in S$ every equivalence class $\sim_s$ has cardinality $\aleph_0$.

We still need to show that $\delta$ is well-defined, i.e. if we have to show the following for any $x,y\in\Delta$: if there are reduced galleries $x_0=x,x_1,\ldots, x_k=y$ and $y_0=x,y_1,\ldots, y_m=y$ of type $(r_1,r_2,\ldots,r_k)$ and $(s_1,\ldots s_m)$, respectively, then in $W$ we have $r_1r_2\ldots r_k=s_1\ldots s_m$. Equivalentely, we will show the following, which completes the proof of Theorem \[t:building\]:

[**Claim:**]{} There is no reduced gallery $a_0,a_1,\ldots, a_k=a_0$ for $k>0$ in $M_n^0$.

*Proof of Claim.* Suppose otherwise. Let $a_0,a_1,\ldots, a_k=a_0$ be a reduced gallery of type $(r_1,\ldots r_k)$ for some $k>0$. Note that the flags $a_i$ and $a_{i+1}$ contain the same vertex of type $V_j$ as long as $r_i\neq t_j$.

Now consider the sequence of vertices of type $V_n$ and $V_{n-1}$ occurring in this gallery. Since $V_n\cup V_{n-1}$ contains no cycles, the sequence of vertices of type $V_n$ and $V_{n-1}$ occurring in this gallery will be of the form $$\label{eq:gallery}
  (x_1,y_1,x_2,y_2,\ldots ,x_i,y_i,x_i,y_{i-1},\ldots y_1,x_1)$$ with $x_i\in V_n,y_i\in V_{n-1}$ and $x_i$ a neighbour of $y_i$ and $y_{i-1}$ in the original graph. This implies that at some place in the gallery type there are two occurrences of $t_n$ which are not separated by an occurrence of $t_{n-1}$ (or conversely). Since $t_n$ commutes with all $t_i$ for $i\neq n-1$ and the word $r_1\ldots r_k$ is reduced, there are two occurrences of $t_{n-1}$ which are not separated by an occurrence of $t_n$, say $r_j,r_{j+m}=t_{n-1}$ with $r_{j+1},\ldots, r_{j+m-1}\neq t_n$.

We now consider the gallery $a_j,\ldots a_{j+m}$ of type $(r_j=t_{n-1},r_{j+1},\ldots, r_{j+m}=t_{n-1})$. Notice that by (\[eq:gallery\]), the flags $a_j$ and $a_{j+m}$ have the same $V_n$ and the same $V_{n-1}$ vertex. Since $M_n^0$ does not contain any $E_{n-1}$-cycles, the sequence of $V_{n-1}$- and $V_{n-2}$-vertices appearing in this sequence must again be of the same form as in (\[eq:gallery\]). Exactly as before we find two occurrences[^1] of $t_{n-2}$ in the gallery type of $a_j,\ldots a_{j+m}$ which are not separated by an occurrence of $t_{n-1}$. Continuiung in this way, we eventually find two occurrences of $t_1$ which are not separated by any $t_i$. Since $t_1^2=1$ this contradicts the assumption that the gallery be reduced.

The proof shows in fact the following:

A model of $T_n$ is a building of type $A_{\infty,n+1}$ if and only if it is $E_i$-connected for all $i$ and every vertex is contained in a maximal flag.

The building $M_n^0$ is the prime model of $T_n$

To see that $M_n^0$ is the prime model of $T_n$ note that for any flags $C_1,C_2\in M_n^0$ and gallery $C_1,=x_0,\ldots ,x_k=C_2$ the set of vertices occuring in this gallery is $E_i$-connected for all $i$. Hence by Remark \[r:qftp\] its type is determined by the quantifier-free type.

Thus, given a maximal flag $M$ in any model of $T_n$ and a maximal flag $c_0$ of $M_n^0$ we can embed $M_n^0$ into $M$ by moving along the galleries of $M_n^0$.

Ranks and types {#sec:types}
===============

Recall that for vertices $x,y\in M_n^0$ with $x\in V_i,y\in V_j$ the *Weyl-distance* $\delta(x,y)$ equals $w\in W$ if there are flags $C_1,C_2$ containing $x,y$, respectively, with $\delta(C_1,C_2)=w'$ and such that $w$ is the shortest representative of the double coset $\langle t_k\colon k\neq i\rangle^W w'\langle t_k\colon k\neq j\rangle^W$ (where as usual $\langle X\rangle^W$ denotes the subgroup of $W$ generated by $X$).

The following is clear:

\[p:qe\] The theory $T_n$ has quantifier elimination in a language containing predicates $\delta_w^{i,j}$ for Weyl distances between vertices of type $V_i$ and of type $V_j$.

For any small set $A$ in a large saturated model we have the following kinds of regular types:

1.  $\operatorname{tp}(a/A)$ where $a\in V_i$ is not connected to any element in $\operatorname{acl}(A)$

2.  $\operatorname{tp}(a/A)$ where $a\in V_i$ is incident with some $b\in\operatorname{acl}(A)\cap V_j$ but not connected in $R(b)$ to any vertex in $\operatorname{acl}(A)\cap R(b)$.

3.  $\operatorname{tp}(a/A)$ where $a\in V_i$ is incident with some $x,y\in \operatorname{acl}(A)$ such that $(x,a,y)$ is a flag with $x\in V_k,y\in V_j$; and as a special case of this we have

4.  $\operatorname{tp}(a/A)$ where $a\in V_i$ has neighbours $x,y\in \operatorname{acl}(A)$ such that $(x,a,y)$ is a (necessarily dense) flag.

By quantifier elimination any of these descriptions determines a complete type. Using the description of forking in Corollary \[c:forking\] one sees easily that each of these types is regular and trivial.

Clearly, any type in (IV) has $U$-rank $1$ and in fact Morley rank $1$ by quantifier elimination. It also follows easily that $\operatorname{MR}(a/A)=\omega^n$ if $\operatorname{tp}(a/A)$ is as in (I). In case (II) we find that $\operatorname{MR}(a/A)=\omega^{n-j-1}$ or $\operatorname{MR}(a/A)=\omega^{j-1}$ depending on whether or not $i<j$. In case (II) we have $\operatorname{MR}(a/A)=\omega^{|k-j|-2}$.

Just as in [@BP] we obtain:

Any regular type in $T_n$ is non-orthogonal to a type as in (I) or as in (IV).

Let $p=\operatorname{tp}(b/\operatorname{acl}(B))$. If $b$ is not connected to $\operatorname{acl}(B)$, then $p$ is as in (I), so we may assume that $\operatorname{proj}(b/B)=C\neq\emptyset$. Let $a$ be a vertex on a short path from $b$ to $C$ incident with an element of $C$. Then by Corollary \[c:forking\] we see that $p$ is non-orthogonal to $\operatorname{tp}(a/C)$.

[**Acknowledgement:**]{} I would like to thank Linus Kramer, in particular for providing reference [@HP].

[WWW]{} A. Baudisch, A. Pillay, A free pseudospace. J. Symbolic Logic 65 (2000), no. 1, 443 – 460.

Th. Grundhöfer, Basics on buildings. Tits buildings and the model theory of groups (Würzburg, 2000), 1 – 21, London Math. Soc. Lecture Note Ser., 291, Cambridge Univ. Press, Cambridge, 2002.

F. Haglund, F. Paulin, Constructions arborescentes d’immeubles. Math. Ann. 325 (2003), no. 1, 137 – 164.

K. Tent, M. Ziegler, A Course in Model theory, to appear in ASL Lecture Notes in Logic, Cambridge University Press, 2012.

K. Tent, M. Ziegler, On the isometry group of the Urysohn space, submitted.

[^1]: If $t_{n-2}$ does not occur in the type of the gallery, this would contradict the assumption that the type is reduced since $t_{n-1}$ commutes with all $t_i$ for $i\neq n,n-2$.
