---
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 determine the orthogonality classes of regular types. 

The construction given here is quite similar to the one given by Evans in [@evans] for a stable theory which is $n$-ample for all $n$, but does not interpret an infinite group.  In contrast to the examples constructed by Evans, our theory is trivial and no infinite group is definable.  Baudisch, Pizarro and Ziegler informed me that they also showed that the examples in [@BP] are not $3$-ample.  In thank Anand Pillay for pointing out a mistake in an earlier version of this paper and Linus Kramer, in particular for providing reference [@HP]. 

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 $E$. If $x\in V_i$ we also say that $x$ is of level $i$.

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 $k$ not contained in $R(a)$, then there is an $E_{n-1}$-path from $b$ to $b'$ in $R(a)$ of length at most $k-1$.

    2.   If $a$ is a vertex of type $V_0$ and $\gamma=(a,b,\ldots , b',a)$ is a simple cycle of length $k$ not contained in $R(a)$, then there is an $E_2$-path from $b$ to $b'$ in $R(a)$ of length at most $k-1$.

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 $\K_n$ 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 a simple cycle of length $k$ not contained in $R(a)$, then there is some $E_{i-1}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$.

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 a simple cycle of length $k$ not contained in $R(a)$, then there is some $E_{i+2}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$.

Let $A$ be a finite $L_n$-graph.  The following extensions are called $1$-point 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 $1$-point strong extensions. 

We next show that $(\K,\leq)$ has the amalgamation property for strong extensions.  This will be enough to obtain a strong limit which is well-defined up to automorphism (see [@ziegler]). 

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 $1$-point 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 strong Fraïssé limit $M_n$. 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. 

\[l:K-sat\] An $L_n$-structure $M$ is an $\omega$-saturated model of $T_n$ 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 $B$ is an $1$-point strong extension of $A$ and by $\omega$-saturation it is easy to see that $B$ can be imbedded over $A$ into $M$. Conversely assume that $M$ is $\K_n$-saturated.  Since $M$ is partially isomorphic to $M_n$, it is a model of $T_n$. Choose an $\omega$-saturated $M'\equiv M$. Then by the above $M'$ is $\K_n$-saturated.  So $M'$ and $M$ are partially isomorphic, which implies that $M$ is $\omega$-saturated. 

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.) 

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. 

\[c:acl2\] The algebraic closure $\operatorname{acl}(A)$ contains the intersection of all nice sets containing $A$. 

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. 

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

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

2.   any path $(x_{i-1},x_i,z_1,\ldots,z_t, x_{i+k},x_{i+k+1})$ contained in $V_j\cup\ldots \cup V_{j+m-1}$ or in $V_{j+1}\cup\ldots \cup V_{j+m}$ is reduced if $(x_i,z_1,\ldots,z_t, x_{i+k})$ is reduced. 

Note that the definition immediately implies 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. 

Using the fact that $M_n$ is $\omega$-saturated we can now describe the algebraic closure:

\[l:acl\] 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. 

We have in fact $\operatorname{acl}(ab)=\operatorname{dcl}(ab)$.

If there is no reduced path between $a$ and $b$ changing direction in $c$, then $c$ has infinitely many conjugates over $ab$, hence $c\notin\operatorname{acl}(ab)$. So suppose there is a reduced path from $a$ to $b$ changing direction in $c$. Let $C_{a,b}$ the set of all vertices $y$ such that there is a reduced path $(a,\ldots, x,y,z,\ldots b)$ changing direction in $y$ and such that for $x,z\in V_j$ are not connected by both an $E_j$ and an $E_{j+1}$-path. 

We claim that $C_{a,b}$ is a finite set.  Let $\gamma=(a,\ldots b)$ be a reduced path.  For any $y\in  C_{a,b}\setminus\gamma$ let $\gamma_y=(a,\ldots, x,y,x,\ldots b) $ be a reduced path witnessing that $y\in C_{a,b}$. Composing suitable pieces of $\gamma$ and $\gamma_y$ we obtain a simple cycle $\gamma_y``$ changing direction in $y$. Since $y\in C_{a,b}$ we have $\gamma_{y}''\subset R(y)$ by $(\Sigma 4)$. For $y_1\neq y_2\in V_j\cap C_{a,b}$ the paths $\gamma_{y_1}``\cap\gamma$ and $\gamma_{y_2}``\cap\gamma$ must be disjoined since otherwise we obtain a simple cycle changing direction in $y_1$ and $y_2$, but not contained in $R(y_1)$ which would contradict $y_1\in C_{a,b}$.

Clearly $C_{a,b}$ is invariant under all automorphisms fixing $a,b$ and so $C_{a,b}\subseteq \operatorname{acl}(ab)$.

Now consider a reduced path $\gamma$ from $a$ to $b$ changing direction in $c$. We may inductively assume that $\gamma\cap\operatorname{acl}(ab)=\{a,b\}$ since otherwise we may replace $a,b$ by some $a',b'\in\operatorname{acl}(ab)\cap \gamma$ and consider the piece of $\gamma$ containing $c$. In particular $\gamma\cap\ C_{a,b}=\emptyset$.

It is therefore sufficient to prove that any reduced path $\gamma=(a,\ldots, b)$ with $\gamma\cap \operatorname{acl}(ab)=\{a,b\}$ is a flag.  We do induction on the number of levels involved.  Clearly, if $\gamma\subseteq V_{j}\cup V_{j+1}$ for some $j$, then $\gamma\subseteq\operatorname{acl}(ab)$ since such a path is unique.  Hence $\gamma=(a,b)$. Now suppose that $\gamma=(a,a_1,\ldots b)\subseteq V_{j}\cup\ldots\cup V_{j+m}$. We claim that $a\in V_j\cup V_{j+m}$: otherwise we may replace all paths $(x,c,y)$ where $\gamma$ changes direction by $E_k$ paths for appropriate $k$ and reduce the number of levels of $\gamma$ (since $\gamma\cap C_{a,b}=\emptyset$).  By induction this new path is a flag, which clearly is impossible.  Hence $a\in V_j\cup V_{j+m}$ and we may apply the same consideration to the subpath of $\gamma$ starting at $a_1$. By induction $(a_1,\ldots b)$ and hence $\gamma$ are flags. 

In Section \[sec:building\] we will see that in the prime model the algebraic closure will be described by reduced words in the Coxeter group associated to the building. 

\[l:treeacl\] 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 Lemma \[l:acl\] 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 symmetry we may assume that $d\notin V_{i+k}$. We may then replace any element $x\in V_{i+k}$ in the interior of $\gamma_1$ and $\gamma_2$ by a reduced path in $R(x)$ between its neighbours.  If $a,b,c\in V_{i+k}$, then we may extend $\gamma_1,\gamma_2$ by vertices in $V_{i+k-1}$ to reduced paths containing $a,b,c$ in its interior and also replace $a,b,c$. We may thus replace the paths $(b,\ldots,d,\ldots, c)$ and $(a,\ldots g,\ldots, d)$ by reduced paths $(b_1,\ldots,d, \ldots,c_1)$ and $(a_1,\ldots d)$ contained in $V_i\cup\ldots\cup V_{i+k-1}$.

By induction assumption, at least one of $(a_1,\ldots d,\ldots b_1)$ and $(a_1,\ldots d,\ldots c_1)$ is reduced, and replacing the new pieces of the path by the old ones, this path remains reduced and changes direction in $g$.

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 Lemma \[l:acl\] $x_i,y_j$ 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 given by Pillay in [@Pi]. 

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 $i=0,\ldots n-1$.

\[r:evans\_ample\] In [@evans], Evans requires the slightly more natural condition

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

\[T:n-ample\] The theory $T_n$ is $n$-ample (in the sense of Evans’ defintion) 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 \[l:acl\] 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 Lemma \[l:acl\] 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:

-   For any $a\in A$ 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 $1$-point strong extensions of $A$:

1.   add a vertex of type $V_0$ or $V_n$ 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 $A\subset V_0\cup V_n, |A|\leq 1$, 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 $1$-point 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 $1$-point 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 $T_n$. Furthermore $M^0_n$ 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. 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), (II), or (III). 

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 $\operatorname{tp}(a/C)$ and $\operatorname{tp}(a/C)$ is of type (II) or (III). 

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[^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 $i\neq n,n-2$.
