[columns=1,title=List of notations, intoc]

# Normal Subgroup Theorem for groups acting on $\tilde{A}_{2}$-buildings

Uri Bader, Alex Furman, Jean Lécureux

###### Abstract

Let $\Gamma$ be a group acting on with finite stabilizers and finite
fundamental domain on a building of type $\tilde{A}_{2}$. We prove that any
non-trivial normal subgroup of $\Gamma$ is of finite index in $\Gamma$.

###### Contents

  1. 1 Spaces of maps as measured spaces
    1. 1.1 Prouniform measures
    2. 1.2 Pushforwards and martingales
  2. 2 Buildings of type $\tilde{A}_{2}$
    1. 2.1 Generalities
    2. 2.2 Boundaries
    3. 2.3 Wall-trees and projectivities
  3. 3 Symmetricity of buildings
    1. 3.1 Convex subset of wall trees
      1. 3.1.1 Crazy diamonds
      2. 3.1.2 Diamond extensions
    2. 3.2 Flows and measures
      1. 3.2.1 Harmonic measures and the Cartan Flow
      2. 3.2.2 Martingales
      3. 3.2.3 Singular flow
  4. 4 Approximation of the singular flow
    1. 4.1 The argument for the tree
      1. 4.1.1 A recurrence argument
      2. 4.1.2 Martingales and Lebesgue differentiation
      3. 4.1.3 The detecting flow
    2. 4.2 Singular flow and contracting sequences
    3. 4.3 The detecting flow on the building
    4. 4.4 Contracting properties
    5. 4.5 Dynamics on the boundary
  5. 5 Subalgebras of $L^{\infty}(\Delta)$
    1. 5.1 Four algebras
    2. 5.2 Radial functions
    3. 5.3 Conclusion

## Introduction

The main examples of affine buildings come from Bruhat-Tits theory: if $k$ is
a local field, and $\mathbf{G}$ a $k$-simple algebraic group, then Bruhat and
Tits associated to $\mathbf{G}$ is an affine building on which $\mathbf{G}(k)$
acts. In particular, cocompact lattices in $\mathbf{G}(k)$ also act on $X$,
properly and cocompactly. In higher rank, such lattices are all arithmetic, by
the celebrated work of Margulis and Venkataramana [Margulis, Venka]. In
particular, their algebraic structure is fairly well understood. For example,
one can produce many finite index normal subgroups by taking congruence
kernels. To clarify the terminology, we say that a group $\Gamma$ acting on a
simplicial complex $X$ is a _cocompact_ (or uniform) _lattice_ in $X$ if it
acts with finite stabilizer and a compact fundamental domain.

A fundamental result of Tits (see the last corollary of [Tits_cours84] as well
as the book [WeissBook]) is that all irreducible affine buildings of dimension
at least 3 are Bruhat-Tits buildings. On the other hand, buildings of
dimension 1 are just trees, and their cocompact lattices are virtually free.

The remaining case of buildings of dimension 2 turns out to be quite
interesting. First, there are reducible buildings of dimension 2, namely,
product of trees. In that case, the celebrated work of Burger and Mozes
[BurgerMozes_lattices] produces examples of lattices in product of two trees
which are simple (and in particular, non-linear). Then, there are irreducible
buildings of dimension 2, of three possible type ($\tilde{A}_{2}$,
$\tilde{C}_{2}$ or $\tilde{G}_{2}$), buildings of type $\tilde{A}_{2}$ being
the simplest and most studied examples. In this setting, it is already
nontrivial to produce examples with cocompact lattices which are not Bruhat-
Tits buildings. In the sequel, we will call these buildings _exotic_. The
first construction appeared in [Ronan_triangle] and [Tits_Andrews], where
Ronan and Tits managed to construct the first examples of such buildings.
There have been many other constructions since then. We refer to [BCL, §1.3]
for a more thorough treatment of the literature. In particular, it has been
proved that there exist lattices in buildings of arbitrarily large thickness;
in some cases (panel-regular lattices) it has even been proven that most
buildings are not Bruhat-Tits (this was conjectured in [Witzel] and proved in
[Radu]).

Another potential source of examples are _Galois lattices_ : starting from a
Bruhat-Tits building $X$, associated to an algebraic group $\mathbf{G}(k)$,
the full automorphism group of $X$ is not $\mathbf{G}(k)$, but an extension of
$\operatorname{Aut}(\mathbf{G})(k)$ by a subgroup of $\operatorname{Aut}(k)$,
which is infinite in positive characteristic (see [BorelTits] and [BCL,
Proposition 9.1]). Therefore there may exist a lattice $\Gamma$ in
$\operatorname{Aut}(X)$ which has infinite image in $\operatorname{Aut}(k)$:
this is what we call a Galois lattice. So far it is not known whether Galois
lattices exist or not in higher rank.

These "exotic" buildings of type $\tilde{A}_{2}$ and their lattices remain
largely mysterious, although they have been studied by many authors. For
example, there have been a large amount of work devoted to proving property
(T) and its variations, starting with the paper of Cartwright-Młotkowski-
Steger [CartMlotkT], then followed by Pansu [Pansu] and Żuk’s criterion [Zuk]
; more recently it was proved that these lattices also have Lafforgue’s strong
property (T) [StrongT].

In [BCL] the authors started to study these lattices with a different point of
view, namely, by adapting and applying Margulis’ methods about lattices in
higher rank simple Lie groups. It was proven there that cocompact lattices in
exotic buildings, as well as Galois lattices, are fairly different from their
arithmetic counterparts: they are always non-linear. This paper can be seen as
a continuation of this work. Indeed, we are able to prove the Normal Subgroup
Theorem:

######  Theorem 0.1.

Let $\Gamma$ be a cocompact lattice of a building of type $\tilde{A}_{2}$.
Then any non-trivial normal subgroup of $\Gamma$ is of finite index.

In the case when the building is Bruhat-Tits, and $\Gamma<\mathbf{G}(k)$, then
this theorem is due to Margulis [Margulis]. Therefore our main contribution is
about exotic buildings, and possibly also to Galois lattices. This theorem was
announced by Shalom and Steger in 2008, but the proofs have not appeared in
print, and we do not know how large is the overlap between our approach and
theirs. Our motivation for writing this proof is two-fold. First, it is a
first step towards proving stronger results: the Stuck-Zimmer theorem
[StuckZimmer] or character rigidity, in the spirit of [Peterson].

The second motivation stems from the following corollary.

######  Corollary 0.2.

Let $\Gamma$ be a cocompact lattice of a building of type $\tilde{A}_{2}$.
Then either $\Gamma$ is virtually simple, or it is residually finite.

###### Proof.

If $\Gamma$ is not residually finite, then the intersection $N$ of all finite-
index subgroups of $\Gamma$ is not trivial. But $N$ is a normal subgroup of
$\Gamma$, so it is of finite index, and therefore is also a cocompact lattice
of the same building. Applying Theorem 0.1 to $N$, we get that every non-
trivial normal subgroup of $N$ is of finite index in $N$, and therefore also
in $\Gamma$, so that it contains $N$. Hence $N$ is a finite index simple
subgroup of $\Gamma$. ∎

If $\Gamma$ is a lattice in an algebraic group over a local field, then (as
mentioned above) it is clear that $\Gamma$ is residually finite. On the other
hand, if $\Gamma$ is a cocompact lattice in an exotic building, then the
situation is very much unclear. It was conjectured in [BCL, Conjecture 1.5]
that the first conclusion always holds ($\Gamma$ is virtually simple), but so
far there have been very few progress in that direction. However, recently,
Thomas Titz Mite and Stefan Witzel have been able to produce cocompact
lattices in buildings of type $\tilde{C}_{2}$ which are not residually finite
[MiteWitzel]. Although the proof needs to be adapted in non-trivial ways, it
is very likely that the analogue of Theorem 0.1 also holds for $\tilde{C}_{2}$
buildings, thus providing new examples of infinite simple groups.

The main steps of the proof of Theorem 0.1 follow the beautiful strategy of
Margulis. If $N$ is a non-trivial normal subgroup of $\Gamma$, then we first
prove that $\Gamma/N$ has property (T) (we get this for free as $\Gamma$
itself has property (T)), and then that $\Gamma/N$ is amenable, which is the
hard part. To do that, we introduce the boundary $\Delta$ of the building,
which is an amenable $\Gamma$-space. The space $\Delta$ is the set of chambers
of a spherical building (of rank 2, hence of dimension 1, i.e a graph), and is
equipped with two $\Gamma$-equivariant maps $\Delta\to\Delta_{+}$ and
$\Delta\to\Delta_{-}$, where $\Delta_{+}$ and $\Delta_{-}$ are the two sets of
vertices of different types. The space $\Delta$ is endowed with a natural
measure class, which is $\operatorname{Aut}(X)$-invariant. As in Margulis’
proof, the amenability of $\Gamma/N$ will follow from a Factor Theorem. In our
case this Factor Theorem translates as follows.

######  Theorem 0.3.

Any $\Gamma$-equivariant measurable factor $\Delta\to Z$ is (measurably)
isomorphic to either $\Delta\to\Delta_{+}$, $\Delta\to\Delta_{-}$, the
identity map or a point.

The proof of this Theorem is our main contribution. The general strategy to
prove this theorem is to find enough elements in $\Gamma$ with some kind of
contracting properties: namely, contracting almost all of $\Delta$ to the set
of chambers adjacent to a given vertex $u\in\Delta_{-}$ (the _residue_
$\operatorname{Res}(u)$ of $u$). In fact, what we prove is that one can
approximate a special subgroup of $\operatorname{Aut}(\operatorname{Res}(u))$
called the _projectivity group_. This group is a classical object of study in
incidence geometries and possess a very rich structure, even when the building
has few automorphisms. In particular, it is always 3-transitive on
$\operatorname{Res}(u)$.

The statement of Theorem 0.3 refers to a measure class on $\Delta$, which is
naturally defined. Note also that the strategy sketched in the previous
paragraph naturally gives some topological convergence, and therefore it
requires some more work to get a statement about the measure class. In the
classical case, this is obtained through a non-trivial version of the Lebesgue
Density Theorem [Margulis, IV.1]. Furthermore, even in the classical case, the
proof uses various actions on homogeneous spaces, with their measured
structure.

In our setting, these homogeneous spaces will be replaced by some natural sets
of embeddings: for example, the set of embeddings of an apartment into the
building. It is not obvious in general how to construct natural measures with
good properties on such spaces. In order to do so, we were led to introduce a
fairly general formalism which explains how and when one can construct
_prouniform_ measures on the set of embeddings of a simplicial complex into
another, under some condition that we call _P-symmetricity_. We then check
that this condition holds for all the spaces that we need. Finally, we found
that it was more suited to our setting to use the Martingale Convergence
Theorem rather in place of the Lebesgue Density Theorem. Furthermore, in order
to attach the kind of convergence we obtain to the group action, we were led
to introduce yet another space, that we call the _detecting flow_ , which can
be thought as an kind of geodesic flow (or Cartan flow in our case) with
additionnal "antennas".

Structure of the paper. The first section (after this introduction) explains
the construction of prouniform measures on sets of embeddings between
simplicial complexes, and can be read completely independently of the paper.
The second section describes quickly the background on buildings which is
necessary to us, and then in §3 we check that the assumptions needed in §1 are
indeed true in the setting we want. It is also the occasion to define the
various spaces and flows that we will use in the course of the proof. The
proof itself starts in Section 4, where we explain how to produce the elements
of $\Gamma$ with the required dynamical properties. Finally, in §5, we
conclude the proof of Theorem 0.3 and 0.1.

In this work we were led to introduce many different spaces. To help the
reader keep track of everything, we also included a list of notations at the
very end of the paper.

Acknowledgment: We are thankful to Pierre-Emmanuel Caprace and Stefan Witzel
for various discussions and encouragements.

##  1 Spaces of maps as measured spaces

###  1.1 Prouniform measures

We consider here connected simplicial complexes. In this section, for a
connected simplicial complex $Y$ we denote by $Y^{k}$ the set of its
$k$-dimensional simplices. In particular, $Y^{0}$ is its set of vertices.
$Y^{0}$ is endowed with the combinatorial metric given by the 1-skeleton, that
is the graph $(Y^{0},Y^{1})$. For two connected simplical complexes $Y$ and
$Y^{\prime}$, by isometry from $Y^{\prime}$ to $Y$ we mean a simplicial map
which induces an isometry $Y^{\prime 0}\to Y^{0}$ and such that every simplex
in $Y$ whose boundary is in the image of $Y^{\prime}$ is itself in the image
of $Y^{\prime}$. We denote by $\text{Isom}(Y,X)$ the set of all such
isometries. If $Y^{\prime}$ is a subsimplicial complex of $Y$ and the
inclusion is an isometry we say that $Y^{\prime}$ is convex in $Y$.

Throughout this section we fix a locally finite connected simplical complex
$X$. Given any finite simplicial complex $Y$, we endow the countable set
$(X^{0})^{Y^{0}}$ with the counting measure and let $m_{Y}$ be its restriction
to $\text{Isom}(Y,X)$. Typically $(\text{Isom}(Y,X),m_{Y})$ is an infinite
measure space, but if $Y$ is connected, $Y^{\prime}$ is nonempty and
$i:Y^{\prime}\to Y$ is an isometry, then the map
$i^{*}:(\text{Isom}(Y,X),m_{Y})\to(\text{Isom}(Y^{\prime},X),m_{Y^{\prime}})$
given by $\beta\mapsto\beta\circ i$ is a finite measure extension, that is
there exits a disintegration of $m_{Y}$ with respect to $m_{Y^{\prime}}$ with
finite measured fibers. Indeed, given an isometry $\alpha:Y\to X$, the set

$\\{\beta:Y\to X\mid~{}\beta\mbox{ is an isometry and }\alpha=\beta\circ
i\\},$

is finite, as $Y$ is connected and $X$ is locally finite, and for every
$U\subset\text{Isom}(Y,X)$,

$m_{Y}(U)=\int_{\text{Isom}(Y^{\prime},X)}|\\{\beta\in U\mid~{}\beta\mbox{ is
an isometry and }\alpha=\beta\circ i\\}|dm_{Y^{\prime}}(\alpha).$

######  Definition 1.1.

Given a property of connected finite simplicial complexes $P$, we say that $X$
is $P$-symmetric if it satisfies the following property:

  * •

For every pair of nonempty finite connected $P$ simplicial complexes
$Y^{\prime},Y$ and an isometry $i:Y^{\prime}\to Y$, given an isometry
$\alpha:Y^{\prime}\to X$ the size of the set

$\\{\beta:Y\to X\mid~{}\beta\mbox{ is an isometry and }\alpha=\beta\circ i\\}$

is positive and independent of $\alpha$.

In case $X$ is $P$-symmetric and $Y,Y^{\prime}$ and $i$ are as above, we
denote the size of the above set by $[Y:Y^{\prime}]_{i}$ or by
$[Y:Y^{\prime}]$, in case $Y^{\prime}\subset Y$ is a convex subset and $i$ is
the inclusion.

######  Example 1.2.

Let $P$ be the property tree of degrees bounded by $d$. Then a $d$-regular
tree is $P$-symmetric.

We fix for the rest of the section a property of finite connected simplicial
complexes $P$. We assume $P$ is such that finite segments (isomorphic to a
convex subset of $\mathbf{Z}$), and in particular points, satisfy $P$. We also
assume that $P$ is stable by taking convex subcomplexes.

We also assume that our fixed simplical complex $X$ is $P$-symmetric.

Note that for a pair of finite connected simplicial complexes $Y^{\prime},Y$
both having $P$ and an isometry $i:Y^{\prime}\to Y$, the push forward of
$m_{Y}$ under the restriction map
$i^{*}:\text{Isom}(Y,X)\to\text{Isom}(Y^{\prime},X)$ satisfies

$(i^{*})_{*}(m_{Y})=[Y:Y^{\prime}]_{i}\cdot m_{Y^{\prime}}.$ (1)

######  Lemma 1.3.

Let $Y$ be a finite connected simplicial complexes satisfying $P$.

  * •

For any isometry $i:Y\to Y$, $[Y,Y]_{i}=1$.

  * •

For finite connected $P$ simplicial complexes $Y^{\prime}$ and
$Y^{\prime\prime}$ and isometries $i:Y^{\prime\prime}\to Y^{\prime}$,
$j:Y^{\prime}\to Y$ one has

$[Y,Y^{\prime\prime}]_{j\circ
i}=[Y,Y^{\prime}]_{j}\cdot[Y^{\prime},Y^{\prime\prime}]_{i}.$

  * •

For every pair of vertices $y,y^{\prime}\in Y_{0}$,
$[Y,\\{y\\}]=[Y,\\{y^{\prime}\\}]$.

###### Proof.

The proof of the first two claims is straight forward and left to the reader.
In order to see the third one, we let $I\subset Y$ be a geodesic segment with
end points $y,y^{\prime}$ and let $\tau:I\to I$ be the unique non-trivial
isometry. Then, using the first two claims, we have
$[I,\\{y\\}]=1\cdot[I,\\{y\\}]=[I,I]_{\tau}\cdot[I,\\{y\\}]=[I,\\{y^{\prime}\\}]$,
thus

$[Y,\\{y\\}]=[Y,I]\cdot[I,\\{y\\}]=[Y,I]\cdot[I,\\{y^{\prime}\\}]=[Y,\\{y^{\prime}\\}].$

∎

In view of the above lemma, we use the notation $[Y]$ to denote the common
value $[Y,\\{y\\}]$ for $y\in Y_{0}$. We observe that for a pair of finite
connected simplicial complexes $Y^{\prime},Y$ both having $P$ and an isometry
$i:Y^{\prime}\to Y$ we have

$[Y]=[Y,Y^{\prime}]_{i}\cdot[Y^{\prime}].$ (2)

Fix an origin $y_{0}\in Y^{0}$ and $o\in X^{0}$. We define a probability
measure $\mu_{Y}^{o}$ on $\operatorname{Isom}(Y,X)$ by declaring it to be the
uniform probability on the (finite) set
$\operatorname{Isom}(Y,X)^{o}:=\\{\beta\in\operatorname{Isom}(Y,X)\mid\beta(y_{0})=o\\}$.

More generally, if $i:Y^{\prime}\to Y$ is an isometry, and
$\alpha:Y^{\prime}\to X$ is a given isometry, we write
$\mu_{Y,Y^{\prime}}^{\alpha}$ to be the uniform probability on
$\operatorname{Isom}(Y,X)^{\alpha}:=\\{\beta\in\operatorname{Isom}(Y,X)\mid\beta\circ
i=\alpha\\}$. If $Y^{\prime}=\\{y_{0}\\}$ of course we recover the same
construction.

For every finite connected $P$ simplicial complexes $Y$ we define the measure
$\mu_{Y}$ on $\text{Isom}(Y,X)$ by $\mu_{Y}=\sum\limits_{o\in
X_{0}}\mu_{Y}^{o}$.

######  Lemma 1.4.

For every finite connected $P$ simplicial complexes $Y$ we have

$[Y]\cdot\mu_{Y}=m_{Y}.$ (3)

and for every pair of finite connected simplicial complexes $Y^{\prime},Y$
both having $P$ and an isometry $i:Y^{\prime}\to Y$ we have

$(i^{*})_{*}\mu_{Y}=\mu_{Y^{\prime}}$ (4)

###### Proof.

Since $\operatorname{Isom}(Y,X)$ is countable, it suffices to prove Equation
(3) on singletons. Let $f\in\operatorname{Isom}(Y,X)$, by construction we have
$\mu_{Y}(f)=\mu_{Y}^{o}(f)$ where $o=f(y_{0})$, and this number is by
definition $\frac{1}{[Y]}$.

Equation (4) follows by combining Equations (1), (2) and (3). ∎

The following lemma is a simple double counting argument :

######  Lemma 1.5.

Let $Y_{0},Y_{1},Y_{2}$ be finite connected simplicial complexes having $P$,
with isometries $i_{k}:Y_{k}\to Y_{k+1}$, and
$\alpha_{0}\in\operatorname{Isom}(Y_{0},X)$.

Then

$\mu_{Y_{2},Y_{0}}^{\alpha_{0}}=\sum_{\alpha_{1}\in\operatorname{Isom}(Y_{1},X)}\mu_{Y_{1},Y_{0}}^{\alpha_{0}}(\alpha_{1})\cdot\mu_{Y_{2},Y_{1}}^{\alpha_{1}}$

In particular, taking $Y_{0}$ to be a point we get

$\mu_{Y_{2}}^{o}=\sum_{\alpha_{1}\in\operatorname{Isom}(Y_{1},X)}\mu_{Y_{1}}^{o}(\alpha_{1})\cdot\mu_{Y_{2},Y_{1}}^{\alpha_{1}}$

###### Proof.

Clearly it suffices to prove the first equation. Ongce again it suffices to
prove the Lemma on singletons. So let
$f\in\operatorname{Isom}(Y_{2},X)^{\alpha}_{0}$, and let
$\alpha^{\prime}_{1}=f\circ i_{1}$. On the left-hand side,
$\mu_{Y_{2},Y_{0}}^{\alpha_{0}}(f)=\frac{1}{[Y_{2}:Y_{0}]}$. On the right-hand
side, each term of the sum is $0$ except when
$\alpha_{1}=\alpha^{\prime}_{1}$, so that we get
$\mu_{Y_{2},Y_{1}}^{\alpha^{\prime}_{1}}(f)\mu_{Y_{1},Y_{0}}^{\alpha_{0}}(\alpha^{\prime}_{1})$.

But
$\mu_{Y_{1},Y_{0}}^{\alpha_{0}}(\alpha^{\prime}_{1})=\frac{1}{[Y_{1}:Y_{0}]}$
and $\mu_{Y_{2},Y_{1}}^{\alpha^{\prime}_{1}}(f)=\frac{1}{[Y_{2}:Y_{1}]}$ by
definition. The result follows from Lemma 1.3. ∎

######  Definition 1.6.

Given a property of connected finite simplicial complexes $P$, a connected
simplicial complex is said to be ind-$P$ if it could be presented as an
ascending union of finite convex subcomplexes having property $P$.

We let $Y$ be a connected ind-$P$ simplicial complex, and note that in the
category of sets

$\text{Isom}(Y,X)=\varprojlim\text{Isom}(Y^{\prime},X),$

where $Y^{\prime}$ runs over the directe set of finite, convex subcomplex of
$Y$ having property $P$ (with the inclusion), and the map associated to an
inclusion $i_{Y^{\prime},Y^{\prime\prime}}:Y^{\prime}\to Y^{\prime\prime}$ are
the restrictions
$i_{Y^{\prime},Y^{\prime\prime}}^{*}:\operatorname{Isom}(Y^{\prime\prime},X)\to\operatorname{Isom}(Y^{\prime},X)$.

By Equation (4), the map $i_{Y^{\prime},Y^{\prime\prime}}^{*}$ is measure-
preserving. Therefore, using for example Caratheodory’s extension Theorem,
there exists a unique measure $\mu_{Y}$ on $\operatorname{Isom}(Y,X)$ such
that, for every finite convex subcomplex $Y^{\prime}\subset Y$ with property
$P$, the restriction map
$i_{Y,Y^{\prime}}^{*}:\operatorname{Isom}(Y,X)\to\operatorname{Isom}(Y^{\prime},X)$
satisfies $(i_{Y,Y^{\prime}}^{*})_{*}\mu_{Y}=\mu_{Y^{\prime}}$.

In other words, the measure $\mu_{Y}$ on $\text{Isom}(Y,X)$ is defined by
setting

$(\text{Isom}(Y,X),\mu_{Y})=\varprojlim(\text{Isom}(Y^{\prime},X),\mu_{Y^{\prime}}),$

where $Y^{\prime}$ runs over all finite, convex subcomplexes of $Y$ having
property $P$ and the limit is taken in the category of measured spaces.

######  Definition 1.7.

The measure $\mu_{Y}$ is called the _prouniform measure_.

More generally, for a pair of connected ind-$P$ simplicial complexes $W,Y$ and
an isometry $i:W\to Y$, and $\alpha\in\operatorname{Isom}(W,X)$, we can
construct a relative version $\mu_{Y,W}^{\alpha}$ as follows. Let $W^{\prime}$
be a connected ind-$P$ simplicial complex, with isometries $i^{\prime}:W\to
W^{\prime}$, $j:W^{\prime}\to Y$ such that $i=j\circ i^{\prime}$. Assume that
$i^{\prime}(W)$ contains all but finitely many simplices of $W^{\prime}$.

######  Lemma 1.8.

The set
$\operatorname{Isom}(W^{\prime},X)^{\alpha}=\\{\beta\in\operatorname{Isom}(W^{\prime},X)\mid\alpha=\beta\circ
i^{\prime}\\}$ is finite, and its size is independent of $\alpha$.

###### Proof.

We identify below $W$ with the convex subset $i^{\prime}(W)$ of $W^{\prime}$.
Let $W_{1}$ be a finite convex subcomplex of $W^{\prime}$ containing all
simplices of $W^{\prime}\setminus W$ and satisfying $P$. Since $W^{\prime}$ is
ind-$P$, this always exists. Let $\alpha_{1}$ be the restriction of $\alpha$
to $W\cap W_{1}$ (which also satisfies $P$ by assumption). Then the number of
extensions of $\alpha_{1}$ to an isometry $\beta_{1}$ of $W_{1}$ is positive,
finite and independent of $\alpha$. Then, using $P$-symmetry again, one can
extends $\beta_{1}$ in a unique way to an isometry
$\beta^{\prime}_{1}\in\operatorname{Isom}(W^{\prime}_{1},X)^{\alpha}$, for any
$W^{\prime}_{1}\subset W^{\prime}$ finite, having P and containing $W_{1}$,
and therefore to a unique isometry
$\beta\in\operatorname{Isom}(W^{\prime},X)^{\alpha}$. ∎

For $W,W^{\prime}$ as above, we equip
$\operatorname{Isom}(W^{\prime},X)^{\alpha}$ with the uniform measure, which
we denote $\mu_{W^{\prime},X}^{\alpha}$.

More generally, define
$\operatorname{Isom}(Y,X)^{\alpha}=\\{\beta\in\operatorname{Isom}(Y,X)\mid\alpha=\beta\circ
i\\}$. Observe that again, in the category of sets, we have

$\text{Isom}(Y,X)^{\alpha}=\varprojlim\text{Isom}(W^{\prime},X)^{\alpha},$

where $W^{\prime}$ runs over all cofinite complexes containing $W$ as a
subcomplex and being ind-P. We use this observation in order to introduce a
measure, $\mu_{Y,W}^{\alpha}$, on $\text{Isom}(Y,X)^{\alpha}$ by setting

$(\text{Isom}(Y,X)^{\alpha},\mu_{Y,W}^{\alpha})=\varprojlim(\text{Isom}(W^{\prime},X),\mu_{Y,W^{\prime}}^{\alpha}),$

In particular, when $W$ is reduced to a point, we define this way the
"restricted" measure $\mu_{Y}^{o}$ for every $o\in X_{0}$. Note that again we
get

$\mu_{Y}=\sum_{o\in X_{0}}\mu_{Y}^{o}$

######  Definition 1.9.

The measures $\mu_{Y}^{\alpha}$ are called _restricted prouniform measures_.

######  Lemma 1.10.

Let $Y_{0},Y_{1},Y_{2}$ be connected ind-$P$ simplicial complexes, with
isometries $i_{k}:Y_{k}\to Y_{k+1}$, and $\alpha_{0}:Y_{0}\to X$ be an
isometry.

Then

$\mu_{Y_{2},Y_{0}}^{\alpha_{0}}=\int_{\operatorname{Isom}(Y_{1},X)}\mu_{Y_{2},Y_{1}}^{\alpha_{1}}d\mu_{Y_{1},Y_{0}}^{\alpha_{0}}(\alpha_{1})$

###### Proof.

It suffices to prove the equality against every locally constant function, and
therefore against the characteristic function of a set of isometries
determined by a finite convex subcomplex of $Y$. In that case this follows
from Lemma 1.5. ∎

In particular if $Y_{0}$ is a point then

$\mu_{Y_{2}}^{o}=\int_{\operatorname{Isom}(Y_{1},X)}\mu_{Y_{2},Y_{1}}^{\alpha_{1}}d\mu_{Y_{1}}^{o}(\alpha_{1})$

Using the lemma we directly get the following.

######  Corollary 1.11.

Let $Y_{0},Y_{1},Y_{2}$ be connected ind-$P$ simplicial complexes, with
isometries $i_{k}:Y_{k}\to Y_{k+1}$, and $\alpha_{0}:Y_{0}\to X$ be an
isometry.

Then
$(i_{1}^{*})_{*}\mu_{Y_{2},Y_{0}}^{\alpha_{0}}=\mu_{Y_{1},Y_{0}}^{\alpha_{0}}$

Again in the special case when $Y_{0}$ is a point we have that
$(i_{1}^{*})_{*}\mu_{Y_{2}}^{o}=\mu_{Y_{1}}^{o}$.

######  Example 1.12.

Let $X$ be a regular tree, and $Y$ be a half-line $[0,+\infty)$. Then the set
of embeddings of $Y$ into $X$ which send $0$ to $o$ can be identified to the
boundary $\partial_{\infty}T$ ; the measure $\mu_{Y}^{o}$ is then the usual
harmonic measure on the tree.

######  Example 1.13.

Let $X$ be a regular tree, $Y$ be a line and $Y^{\prime}\subset Y$ be a half-
line. Then $\operatorname{Isom}(Y,X)$ is the "geodesic flow" on the tree, and
$\mu_{Y}$ is the well-known Bowen-Margulis measure on this space. And as we
saw before, $\operatorname{Isom}(Y^{\prime},X)$ can be identified to the
boundary of the tree. With this identification in mind, the restriction map is
just the map which associates to a geodesic its endpoint.

###  1.2 Pushforwards and martingales

Finally, let us end this section by some statements about spaces on measures
on prouniform spaces. Let $Y,Y^{\prime}$ be an ind-$P$ simplicial complexes,
and $i:Y^{\prime}\to Y$ be an inclusion. Then the restriction $i_{*}$, denoted
$r$ to avoid to write down too many stars, is a map from
$\operatorname{Isom}(Y,X)\to\operatorname{Isom}(Y^{\prime},X)$. Again, to
avoid cumbersome notations, write $I_{Y}=\operatorname{Isom}(Y,X)$. Then we
get a map $r_{*}:L^{1}(I_{Y},\mu_{Y})\to
L^{1}(I_{Y^{\prime}},\mu_{Y^{\prime}})$, which is just defined as the
pushforward of a finite measure. More concretely, the map $r_{*}$ is an
average of $f$ along the fibers of $r$.

On the other hand, we also get a map $r^{*}:L^{1}(I_{Y^{\prime}})\to
L^{1}(I_{Y})$ (or more generally for any $L^{p}$ space, for $1\leq
p\leq+\infty$), defined by $r^{*}f(z)=f(r(z))$.

The dual of $L^{1}(I_{Y},\mu_{Y})$ is of course $L^{\infty}(I_{Y},\mu_{Y})$,
with duality given by integration along $\mu_{Y}$, which we denote by
$\langle\cdot,\cdot\rangle$. The restriction $r:I_{Y}\to I_{Y^{\prime}}$
defines again a pullback $r^{*}:L^{\infty}(I_{Y^{\prime}})\to
L^{\infty}(I_{Y})$.

We first record the following observation.

######  Lemma 1.14.

For every $\varphi\in L^{1}(Y,\mu_{Y})$ and $f\in
L^{\infty}(Y^{\prime},\mu_{Y^{\prime}})$ we have

$\langle\varphi,r^{*}f\rangle=\langle r_{*}\varphi,f\rangle$

###### Proof.

Write the disintegration $d\mu_{Y}(z)=\mu_{Y}^{r(z)}d\mu_{Y^{\prime}}(r(z))$,
where for every $\alpha\in Y^{\prime}$ the measure $\mu_{Y}^{\alpha}$ is
supported on $r^{-1}(\alpha)$.

Then for every $\alpha\in Y^{\prime}$ we get
$r_{*}\varphi(\alpha)=\int\varphi(x)d\mu_{Y}^{\alpha}(x)$ and

$\displaystyle\langle r_{*}\varphi,f\rangle$
$\displaystyle=\int_{I_{Y^{\prime}}}r_{*}\varphi(\alpha)f(\alpha)d\mu_{Y^{\prime}}(\alpha)$
$\displaystyle=\int_{I_{Y^{\prime}}}f(\alpha)\int_{I_{Y}}\varphi(z)d\mu_{Y}^{\alpha}(z)$
$\displaystyle=\int_{I_{Y}}f(r(z))\varphi(z)d\mu_{Y}(z)$
$\displaystyle=\langle\varphi,r^{*}f\rangle$

∎

Our final remark for this section is that our setup is well-suited for
applications of the Martingale Convergence Theorem.

Indeed, let $Y^{\prime}$ is a convex ind-$P$ subset of $Y$, and let
$(Y_{k})_{k\in\mathbf{N}}$ be an increasing sequence of finite convex subsets
of $Y$ with property $P$, such that $Y=\bigcup_{k\in\mathbf{N}}Y_{k}$. Let
$r_{k}:I_{Y}\to I_{Y_{k}}$ be the restriction map. For
$i\in\operatorname{Isom}(Y,X)$ let $\Omega_{k}(i)=r_{k}^{\textemdash
1}(r_{k}(i))$ be the "$k$-shadow" of $i$: it is the set of embedding which
agree with $i$ on $Y_{k}$. Now let $\mathscr{F}_{k}$ be the $\sigma$-algebra
generated by all the $\Omega_{k}(i)$. Similarly let $\mathscr{F}_{\infty}$ be
the $\sigma$-algebra generated by all the $\Omega_{\infty}(i)=r^{-1}r(i)$
where $r:\operatorname{Isom}(Y,X)\to\operatorname{Isom}(Y^{\prime},X)$ is the
restriction. In other words $\mathscr{F}_{k}$ (resp. $\mathscr{F}_{\infty}$)
is the pullback of the $\sigma$-algebra of $\operatorname{Isom}(Y_{k},X)$
(resp. $\operatorname{Isom}(Y^{\prime},X)$) by $r_{k}$ (resp. $r$).

Then by definition, for $f\in L^{1}(\operatorname{Isom}(Y,X))$, the
conditional expectation $\mathbf{E}(f\mid\mathscr{F}_{k})(i)$ is just
$(r_{k})_{*}f(i)$, which is the average of $f$ on $\Omega_{k}(i)$.

In particular the Martingale Convergence Theorem reads as follows

######  Theorem 1.15.

For every $f\in L^{1}(\operatorname{Isom}(Y,X))$, for almost every
$i\in\operatorname{Isom}(Y,X)$ we have

$\lim_{k}\mathbf{E}(f\mid\mathscr{F}_{k})(i)=\mathbf{E}(f\mid\mathscr{F}_{\infty})(i)$

The convergence also holds in $L^{1}$.

Let us emphasize the following consequence.

######  Lemma 1.16.

With the above setting, fix $\alpha\in\operatorname{Isom}(Y^{\prime},X)$. Then
the space
$\bigcup_{k\in\mathbf{N}}r_{k}^{*}L^{1}(I_{Y_{k}},\mu_{Y_{k}}^{\alpha})$ is
dense in $L^{1}(I_{Y},\mu_{Y}^{\alpha})$ (in the norm topology).

###### Proof.

By the previous discussion, we get that for $\phi\in L^{1}(I_{Y})$ the
sequence $(r_{k}^{*}\phi)_{k\in\mathbf{N}}$ is a martingale, bounded by
$\|\phi\|$. Hence it converges (both a.s. and in $L^{1}$) to $\phi$.

∎

##  2 Buildings of type $\tilde{A}_{2}$

In this section we will explain the facts that we need about $\tilde{A}_{2}$
buildings. We assume some familiarity with the notion, and refer the reader to
the reference book [AbramenkoBrown] for a more thorough treatment. Other
references include [WeissBook] or the paper [BCL] which adopted a similar
point of view and notations. Nevertheless, we will start by recalling a few
basic facts and definitions, also in order to fix the vocabulary and
notations. Since we will introduce a large variety of notation, we made a list
gathering them all by the end of the article.

###  2.1 Generalities

Let $\Sigma$ be the tessellation of the Euclidean plane by equilateral
triangles. This is a _model apartment_. Lines of $\Sigma$ are called _walls_ ,
each wall divides $\Sigma$ into two connected components, which are called
_half-apartments_. Maximal simplices (i.e triangles) are called _alcoves_.

######  Definition 2.1.

A _building of type $\tilde{A}_{2}$_ is a simplicial complex $X$, of dimension
2, covered by copies of $\Sigma$ (called _apartments_), such that

  * •

Every two simplices of $X$ are contained in some apartment

  * •

If $A,A^{\prime}$ are two apartments then there exists an isomorphism $A\to
A^{\prime}$ fixing $A\cap A^{\prime}$ pointwise

  * •

Every edge is contained in at least 3 triangles.

From now on we fix an $\tilde{A}_{2}$ building $X$. We equip it with the
maximal system of apartments: every embedded copy of $\Sigma$ in $X$ is an
apartment. The number of triangles containing a given edge can be proven not
to depend on the edge, and we will assume it is finite. This number will be
written $q+1$, where $q>1$ is the _thickness_ of the building.

We define an alcove (resp. a wall, an half-apartment) in the building as the
image of the image of an alcove (resp. a wall, an half-apartment) by an
embedding of $\Sigma$ in $X$.

The following is [AbramenkoBrown, Theorem 11.52].

######  Theorem 2.2.

Let $X$ be an $\tilde{A}_{2}$ building. Let $C\subset X$ be either convex or
of nonempty interior. Assume that $C$ is isometric to a subset of an
apartment. Then there exists an apartment $A\subset X$ containing $C$.

From this we can deduce the following well-known corollaries (see eg
[AbramenkoBrown, Exercice 5.83a)].)

######  Corollary 2.3.

Let $(C_{n})_{n\in\mathbf{N}}$ be a sequence of convex subsets of $X$, with
$C_{i}\subset C_{i+1}$ for all $i\in\mathbf{N}$. Assume that each $C_{i}$ is
contained in an apartment. Then there exists an apartment containing
$\bigcup_{n\in\mathbf{N}}C_{n}$.

######  Corollary 2.4.

Let $\alpha$ be an half-apartment contained in $X$, and $c$ be an alcove with
on edge on the wall $\partial\alpha$. There exists an apartment containing
$\alpha\cup c$.

The vertices of $\Sigma$ can be colored uniquely (up to renumerotation) by
three colors in such a way that any two adjacent vertices are of a different
colors. These colors are also called _types_. Using the second axiom one can
check that this coloration extends to the whole building, so that each vertex
of $X$ has a given type. An automorphism of $\Sigma$, or of $X$, is called
_type-preserving_ if it preserves the type. If it does not, then it induces a
permutation of the set of types, and it is called _type-rotating_ if this
permutation is cyclic.

Fix an origin vertex $0\in\Sigma$. Then the walls passing through $0$ divide
$\Sigma$ into 6 connected components, which are called _sectors_ or _Weyl
chambers_. We choose one of them and call it the _positive Weyl chamber_ of
$\Sigma$, and denote it by $\Lambda$. For every pairs of vertices
$x,y\in\Sigma$, there is a unique type-rotating permutation which sends $x$ to
$0$ and $y$ to an element of $\Lambda$. This element is called the
_combinatorial distance_ and denoted $\sigma(x,y)$. More generally, if $x,y\in
X$ then there is an apartment containing $x$ and $y$, and we can define
$\sigma(x,y)$ as the combinatorial distance between $x$ and $y$ in this
apartment. This does not depend on the choice of the apartment. For
$\lambda\in\Lambda$, and $x\in X$ a vertex, the set

$V_{\lambda}(x)=\\{y\in X\mid\sigma(x,y)=\lambda\\}$

is the _combinatorial sphere_ of radius $\lambda$ centered at $x$.

The _convex hull_ of a set $F\subset X$ of vertices of $X$ is the convex hull
in the 1-skeleton of $X$. It is denoted $\operatorname{Conv}(F)$. If $F$ is
contained in an apartment, then $\operatorname{Conv}(F)$ is the intersection
of all apartments containing $F$. Note that if $x,y$ are two vertices of $X$
then the isometry class of $\operatorname{Conv}(x,y)$ only depends on
$\sigma(x,y)$.

We define the _length_ of an element $\lambda\in\Lambda$ as the distance
between $o$ and $\lambda$ in the 1-skeleton of $\Sigma$. We denote this length
$\ell(\lambda)$.

###  2.2 Boundaries

Let us now define the boundary of $X$ and its structure as a spherical
building. We have defined a sector in $\Sigma$, and as usual we define a
sector in $X$ as the image of a sector in $\Sigma$ by any embedding in
$\operatorname{Isom}(\Sigma,X)$. A sector is _based_ at some vertex of $X$,
which is the image of $0$ by this embedding. Two sectors are _equivalent_ if
their intersection contains another sector. Equivalence classes of sectors are
called _Weyl chambers_ , or sometimes just chambers. The set of all chambers
is denoted by $\Delta$.

######  Lemma 2.5.

For each Weyl chamber $C\in\Delta$, and vertex $x\in X$, there is a unique
sector in the class of $C$ which is based at $x$. This sector is denoted
$Q(x,C)$.

We define, for $x,z\in X$ two vertices, the set

$\Omega_{x}(z)=\\{C\in\Delta\mid z\in Q(x,C)\\}$

When $z$ varies, these sets form a basis for a topology of $\Delta$. This
topology does not depend on $x$, and turns $\Delta$ into a compact metrizable
space. The group $\operatorname{Aut}(X)$ acts on $\Delta$ by homeomorphisms. A
chamber is in the _boundary_ of some apartment $A$ if it is the class of some
sector in $A$. A basic fact is that any two chambers of $X$ are contained in
the boundary of some common apartment.

A _singular ray_ of $X$ is a half-line which is contained in a wall of $X$.
Two singular rays $\rho_{1},\rho_{2}$ are _equivalent_ if there exists an
apartment which contains two sub-rays $\rho^{\prime}_{1}\subset\rho_{1}$ and
$\rho^{\prime}_{2}\subset\rho_{2}$ and such that $\rho^{\prime}_{1}$ and
$\rho^{\prime}_{2}$ are parallel. The _vertices at infinity_ of $X$ are the
equivalence classes of singular rays. In fact, there are two types of singular
rays, depending on the cyclic permutation of the types of vertices in the ray.
We call the two types $+$ and $-$. The set of vertices at infinity of type $+$
is denoted $\Delta_{+}$, and similarly the set of vertices at infinity of type
$-$ is denoted $\Delta_{-}$. For each $u\in\Delta_{+}\cup\Delta_{-}$, and
every vertex $x\in X$, there is a unique singular ray in the class of $u$
starting from $x$, and we will denote it $[xu)$. Once again, this defines a
topology on $\Delta_{-}$ and $\Delta_{+}$, two vertices at infinity
$u,u^{\prime}$ being close if the rays $[xu)$ and $[xu^{\prime})$ share a long
initial segment.

Each sector is bounded by two singular rays. This descends to the
corresponding equivalence classes and allow us to define two maps
$\operatorname{pr}_{+}:\Delta\to\Delta_{+}$ and
$\operatorname{pr}_{-}:\Delta\to\Delta_{-}$. We also say that a chamber $C$
_contains_ the two vertices at infinity $\operatorname{pr}_{+}(C)$ and
$\operatorname{pr}_{-}(C)$. These two maps are continuous and equivariant with
respect to the group of type-rotating automorphisms of $X$. For
$u\in\Delta_{+}$ (resp. $\Delta_{-}$), the _residue_ of $u$, denoted
$\operatorname{Res}(u)$, is the fiber $\operatorname{pr}_{+}^{-1}(u)$ (resp.
$\operatorname{pr}_{-}^{-1}(u)$).

It is easily checked that this defines a bipartite graph structure, with set
of vertices $\Delta_{+}\cup\Delta_{-}$, and with set of edges $\Delta$ (an
edge $C$ being attached to a vertex $u$ if $C$ contains $u$). This graph has
diameter 3, and is in fact a spherical building of type $A_{2}$. In
particular, for each vertex at infinity $u$, one can define a _projection_
$\operatorname{proj}_{u}:\Delta\to\operatorname{Res}(u)$, which associates to
a chamber $C\in\Delta$ the unique chamber
$\operatorname{proj}_{u}(C)\in\operatorname{Res}(u)$ which is at minimal
distance from $C$. This projection is continuous.

Two chambers $C,C^{\prime}$ are _opposite_ if they are at maximal distance,
that is, at distance 3. We denote $\Delta_{\mathrm{op}}$ the set of pairs
$(C,C^{\prime})$ of opposite chambers. Two vertices $u\in\Delta_{-}$ and
$v\in\Delta_{+}$ are _opposite_ if there exists $C\in\Delta$ containing $u$
and $C^{\prime}\in\Delta$ containing $v$ such that $C$ and $C^{\prime}$ are
opposite. We denote $\Delta_{\pm,\mathrm{op}}$ the set of all pairs
$(u,v)\in\Delta_{-}\times\Delta_{+}$ which are opposite. The sets
$\Delta_{\mathrm{op}}$ and $\Delta_{\pm,\mathrm{op}}$ are open subsets of
$\Delta^{2}$ and $\Delta_{-}\times\Delta_{+}$ respectively.

###  2.3 Wall-trees and projectivities

For $(u,v)\in\Delta_{\pm,\mathrm{op}}$, the _interval_ between $u$ and $v$ is
the union of all apartments containing $u$ and $v$ in their boundary. It is
denoted $I(u,v)$. A _singular line_ from $u$ to $v$ is a wall of such an
apartment consisting of the union of two singular rays $[xu)\cup[xv)$ for some
$x$.

This interval, up to isometry, does not depend on $u$ and $v$, and has the
following structure. If $\ell$, $\ell^{\prime}$ are two singular lines between
$u$ and $v$, then $\ell$, $\ell^{\prime}$ are parallel (in some apartment),
hence there is a well-defined distance $d(\ell,\ell^{\prime})$ (which does not
depend on the apartment chosen). The set of all such geodesic, equipped with
this distance, is in fact a $(q+1)$-regular tree. It is denoted $T_{u,v}$.

The _model wall-tree_ is a model for the simplicial complex $I(u,v)$: as a
CAT(0) space, it is isometric to a product $T\times\mathbf{R}$ (where $T$ is a
$(q+1)$ regular tree), but the simplicial structure does not come from this
product decomposition: if $\gamma$ is a geodesic ray in $T$, then the product
$\gamma\times\mathbf{R}$ is isomorphic to the model apartment $\Sigma$. This
model wall-tree is denoted $\Upsilon$.

The automorphism group of this simplicial complex
$\operatorname{Aut}(\Upsilon)$ is a subgroup of
$\operatorname{Aut}(T)\times\operatorname{Isom}(\mathbf{R})$, hence is endowed
with a projection $p_{T}:\operatorname{Aut}(\Upsilon)\to\operatorname{Aut}(T)$
(which is surjective), and
$p_{\mathbf{R}}:\operatorname{Aut}(\Upsilon)\to\operatorname{Aut}(\mathbf{R})$.
Let $\operatorname{Aut}(\Upsilon)^{0}$ be the index 2 subgroup such that
$p_{\mathbf{R}}(\operatorname{Aut}(\Upsilon)^{0})$ is formed only by
translations. We denote by $S_{\Upsilon}$ the group
$\operatorname{Ker}(p_{T})\cap\operatorname{Aut}(\Upsilon)^{0}$ ; it is a
subgroup isomorphic to $\mathbf{Z}$, which we call the group of _translations_
of $\Upsilon$.

For $u\in\Delta_{-}$ (or in $\Delta_{+}$), one can also define a _panel tree_
, by the following construction: if $\rho,\rho^{\prime}$ are two singular rays
pointing towards $u$, then, up to deleting finite segments, they are parallel
in some apartment, so again there is a well-defined distance
$d(\rho,\rho^{\prime})$. This defines a metric space, which depends on $u$,
and will be denoted $T_{u}$. There is a natural map $T_{u,v}\to T_{u}$,
associating to a line $\ell$ from $u$ to $v$ the class of a ray
$\rho\subset\ell$ pointing towards $u$. This map is in fact an isomorphism, so
that $T_{u}$ is again an $(q+1)$-regular tree.

Let $C$ be a Weyl chamber in the residue of $u$. Any sector $Q$ representing
$C$ is a union of geodesic rays pointing towards $u$, hence defines a subset
of $T_{u}$, which is in fact itself a geodesic ray of $T_{u}$. If $Q^{\prime}$
also represents $C$, then $Q\cap Q^{\prime}$ contains a subsector, therefore
the corresponding geodesic rays in $T_{u}$ have an infinite intersection, so
they define a point in the boundary of $T_{u}$. This defines a natural map
$\phi_{u}:\operatorname{Res}(u)\to\partial_{\infty}T_{u}$.

The following Lemma is well-known, see for example [RemyTrojan, Lemma 4.2] for
a proof.

######  Lemma 2.6.

For every $u\in\Delta_{-}$ or $u\in\Delta_{+}$, the map
$\phi_{u}:\operatorname{Res}(u)\to\partial_{\infty}T_{u}$ is an homeomorphism.

If $(u,v)\in\Delta_{\pm,\mathrm{op}}$ then from the previous discussion we
have isomorphisms $T_{u}\to T_{u,v}$ and $T_{u,v}\to T_{v}$. Their composition
is an isomorphism $T_{u}\to T_{v}$ called a _perspectivity_ and denoted
$[u;v]$. If $w$ is another vertex opposite $u$, then the composition
$[v;w]\circ[u;v]$ is denote $[u;v;w]$, and we can define similar notations
with more vertices.

If we have a sequence of vertices $u_{0},u_{1},\dots,u_{n}$ such that $u_{i}$
is opposite $u_{i+1}$ for every $i$, and $u_{n}$ is opposite $u_{0}$, then we
can define the _projectivity_ $[u=u_{0};u_{1};u_{2};\dots;u_{n};u]$. The
_projectivity group_ is the group generated by such projectivities. Up to
conjugation, it does not depend on the choice of $u$. Choosing an
identification of $T_{u}$ with the $q$-regular tree $T$, this provides an
action of the projectivity group on $T$ (which, again, does not depend on $u$
up to conjugation. Its closure in $\operatorname{Aut}(T)$ (equipped with the
compact open topology) is denoted $\Pi$. By construction, $\Pi$ comes with a
natural action on every tree $T_{u}$ and $T_{u,v}$. Using Lemma 2.6 we see
that $\Pi$ also acts on $\operatorname{Res}(u)$.

The following lemma is classical, see for example [CameronBook, Proposition
2.4]

######  Lemma 2.7.

The natural action of $\Pi$ on $\partial T$ is 3-transitive.

##  3 Symmetricity of buildings

Our goal in this section is to explain how the setup of prouniform measures
developed in §1 can be used in the context of buildings.

###  3.1 Convex subset of wall trees

Our first goal is to prove the following theorem. Let us say that a finite
simplicial complex has property $P_{\Upsilon}$ if it is isometric to a convex
subset of a model wall tree $\Upsilon$ in a $q$-regular $\tilde{A}_{2}$
building.

######  Theorem 3.1.

A building of type $\tilde{A_{2}}$ is $P_{\Upsilon}$-symmetric.

Let $X$ be an $\tilde{A}_{2}$ building. From the constructions of Section 1,
we get the following.

######  Proposition 3.2.

Let $\Upsilon$ be a model wall-tree as in §2.3. Then for every (finite or
infinite) convex subset $Z$ of $\Upsilon$ there exists a measure $\mu_{Z}$ on
the set of simplicial isometries $\operatorname{Isom}(Z,X)$ satisfying

  * •

For $Z=\\{\ast\\}$ this is the counting measure on
$X\simeq\operatorname{Isom}(\\{\ast\\},X)$

  * •

For a simplicial isometry $Z_{1}\to Z_{2}$, the restriction map
$\operatorname{Isom}(Z_{2},X)\to\operatorname{Isom}(Z_{1},X)$ is measure
preserving.

In particular, $\mu_{Z}$ is invariant under $Aut(Z)\times Aut(X).$

This proposition will be used for various spaces $Z$ in §3.2 below.

######  Remark 3.3.

Consider the property $P_{X}$ of being isometric to a convex subset of an
$\tilde{A}_{2}$-building. Then it is not true that an $\tilde{A}_{2}$-building
(even a cocompact one) is necessarily $P_{X}$-symmetric. To construct an
example, remember that there exists an exotic buildings $X$ (with a cocompact
lattice) in which there are two different isometry types of balls of radius 2,
but only one of radius 1. Hence if $B_{1}\subset B_{2}$ is an inclusion of a
ball of radius 1 in a ball of radius 2 (of one of these two types) and
$\varphi:B_{1}\to X$ is an embedding, then depending on $\varphi$ there is
either 0 or some positive number of extensions of $\varphi$ to an embedding of
$B_{2}$.

####  3.1.1 Crazy diamonds

In order to prove Theorem 3.1, we will give a combinatorial description of
convex subsets of wall trees.

######  Definition 3.4.

A _decorated tree_ is a finite connected tree $T$, with two distinguished
vertices $x,y$ and two integers $s,t\in\mathbf{Z}$, such that

  * •

$s-t=d(x,y)\mod 2$.

  * •

For every $z\in T$, $s-t-d(x,z)-d(y,z)\geq 0$.

######  Remark 3.5.

  * •

In what follows only the quantity $s-t$ is of interest. In particular we can
always assume $s,t\geq 0$, or even $t=0$.

  * •

Let $l(z)=s-t-d(y,z)-d(x,z)$. Then $l(z)$ is even for every $z\in T$.

######  Definition 3.6.

Let $(T,x,y,s,t)$ be a decorated tree. For $z\in T$, let $c(z)=s-d(x,z)$ and
$f(z)=t+d(y,z)$. Note that by assumption we have $c(z)-f(z)\geq 0$.

The _crazy diamond_ associated to a decorated tree is the subset of $\Upsilon$
defined as

$Y=Y(T,x,y,s,t)=\\{(z,u)|u\in[f(z),c(z)]\\}.$

The space $Y$ is endowed with a metric as follows : each edge of the tree has
length $\frac{\sqrt{3}}{2}$,the metric on $\mathbf{R}$ is the usual metric,
and the metric on $X$ is the product metric. With this metric, it is
isomorphic to a subset of $\Upsilon$, and in fact it is isometric to a convex
subcomplex of $\Upsilon$. We equip it with the associated tessellation.

######  Example 3.7.

Here is an example. The tree $T$ is the line with 5 vertices drawn at the
bottom. Here we took $s=6$ and $t=-1$.

$y$$x$

The projection $Y\to T$ given by the first coordinate will be denoted
$\operatorname{pr}$.

Let $X$ be a $q$-regular $\tilde{A}_{2}$ building, and let $T$ be a
$(q+1)$-regular tree. Recall that $\Upsilon$ is a model wall-tree of $Y$. It
is equipped with a projection $\pi:\Upsilon\to T$.

######  Definition 3.8.

A crazy diamond is called _degenerate_ if it of one of the following form :

  * •

$Y$ is of the form $Y(\\{x\\},x,x,s,s)$, that is, $Y$ is a point

  * •

$Y$ is of the form $Y(\\{x\\},x,x,s,t)$ with $s-t\geq 0$. In that case we say
that $Y$ is _vertical_.

  * •

$Y$ is of the form $Y(T,x,y,s,t)$ where $T$ is a segment whose extremities are
$x$ and $y$, and $s-t=d(x,y)$, in which case we say that $Y$ is _horizontal_.

######  Proposition 3.9.

Every finite convex subcomplex of $\Upsilon$ is isometric to some crazy
diamond.

Recall that a function on a tree is convex if and only if it is convex in
restriction to every geodesic segment.

######  Lemma 3.10.

Let $T$ be a finite connected tree, and $f:T\to\mathbf{Z}$ be a convex
function such that if $x$ is adjacent to $y$ then $f(x)=f(y)\pm 1$. Then there
exists $t\in\mathbf{Z}$ and $x\in T$ such that $f(z)=t+d(z,x)$ for every
vertex $z\in T$.

###### Proof.

The set of points where $f$ attains its minimum is non-empty (since $T$ is
finite) and convex (by convexity of $f$). Since there are no adjacent vertices
with the same value, $f$ is minimal at exactly one point $x$. Let $f(x)=t$. If
$x^{\prime}$ is adjacent to $x$ then we must have
$f(x^{\prime})=f(x)+1=t+d(x^{\prime},x)$. Now if $z\in T$ then considering the
geodesic segment $x,x_{1},\dots,z$ we see (by convexity) that $f$ must be
increasing along this segment, hence that $f(z)=t+d(z,x)$. ∎

###### Proof of Proposition 3.9.

Let $Y$ be a finite convex subset of $\Upsilon$. Recall
$\pi:\Upsilon\to\mathscr{T}$ is the natural projection, and let $T=\pi(Y)$.
Note that $T$ is a finite, connected subtree of $\mathscr{T}$. Since $Y$ is a
convex subcomplex, $Y$ is completely determined by the sets $\pi^{-1}(x)\cap
Y$, for $x$ a vertex in $T$, which are closed segments.

Let $u\in\partial_{\infty}\Upsilon$ be an endpoint of a geodesic line
$\ell\subset\Upsilon$ such that $\pi(\ell)$ is just a point. Let $h$ be the
horofunction on $X$ based at $u$ (relative to an arbitrary origin
$o\in\Upsilon$). For $x\in T$, define $f(x)=\min\\{2h(p)\mid
p\in\pi^{-1}(x)\cap Y\\}$ and $c(x)=\max\\{2h(p)\mid p\in\pi^{-1}(x)\cap
Y\\}$. Note that by convexity of $Y$, if $x,y\in T$ are adjacent vertices, and
if $p\in\pi^{-1}(x)\cap Y$ is a vertex of $Y$, then $\pi^{-1}(y)\cap Y$
contains a vertex $q$ which is adjacent to $p$, hence such that
$h(q)=h(p)\pm\frac{1}{2}$. Hence, we get that $f(y)=f(x)\pm 1$ and
$c(y)=c(x)\pm 1$. Note furthermore that, by construction of $\Upsilon$, if
$p,q\in\Upsilon$ are vertices, then $2(h(p)-h(q))$ is of the same parity as
$d(\pi(p),\pi(q))$.

If $\ell$ is a geodesic segment in $T$, then $\ell$ is contained in a geodesic
bi-infinite line (still denoted $\ell$) in $\mathscr{T}$. Then
$A=\pi^{-1}(\ell)\subset\Upsilon$ is an apartment of $X$. Hence its
intersection with $Y$ is a convex subset. Since this intersection is precisely
$\\{p\mid f(\pi(p))\leq 2h(p)\leq c(\pi(p))\\}$, it follows that $f$ is convex
and $c$ is concave in restriction to $s$. Since this is true for every
segment, $f$ is convex on $T$. Hence by Lemma 3.10, there exists $y\in T$ and
$t\in\mathbf{Z}$ such that $f(z)=t+d(z,y)$. Similarly, $-c$ is convex, so
there exists $s\in\mathbf{Z}$ and $x\in T$ such that $c(z)=s-d(y,z)$. Note
also that $f(y)-c(x)=t-s$ is of the same parity as $d(y,x)$ by the above
remark.

By construction it is clear that $Y$ is isometric to the crazy diamond
$Y(T,x,y,s,t)$. ∎

######  Remark 3.11.

The same argument also proves that every convex subset of a crazy diamond is
again a crazy diamond.

####  3.1.2 Diamond extensions

Let $Y=Y(T,x,y,s,t)$ be a crazy diamond. In what follows, for $z\in T$, we
note $c(z)=s-d(x,z)$, $f(z)=t+d(y,z)$ and $l(z)=c(z)-f(z)$. If $Y^{\prime}=Y$
the same quantities for $Y^{\prime}$. will be denoted with primes.

Now we are interested in an isometric inclusion $Y\subset Y^{\prime}$ of crazy
diamonds. Note that if $T\subset T^{\prime}$, and $c(z)\leq c^{\prime}(z)$ and
$f(z)\geq f^{\prime}(z)$ for every $z\in T$ then we get that

$\\{(z,u)\in T\times\mathbf{R}|u\in[f(z),c(z)]\\}\subset\\{(z,u)\in
T^{\prime}\times\mathbf{R}|u\in[f^{\prime}(z),c^{\prime}(z)]\\}.$

Hence $Y\subset Y^{\prime}$.

Every embedding is in fact of this form :

######  Lemma 3.12.

Let $Y\subset Y^{\prime}$ be an isometric inclusion of crazy diamonds. Let
$Y^{\prime}=Y(T^{\prime},x^{\prime},y^{\prime},s^{\prime},t^{\prime})$. Then
there exists a subtree $T\subset T^{\prime}$, $x,y\in T$ and
$s,t\in\mathbf{Z}$ such that $Y=Y(T,x,y,s,t)$ and the inclusion $Y\subset
Y^{\prime}$ is as described above.

###### Proof.

Embed $Y^{\prime}$ (and therefore $Y$) into $\Upsilon$. Going through the
proof of Proposition 3.9, we see that $T^{\prime}=\pi(Y^{\prime})$ and
$T=\pi(Y)$, so that clearly $T\subset T^{\prime}$, and by convexity of $Y$,
$T$ is also convex and therefore is a subtree of $T^{\prime}$. Similarly by
the construction given in the proof of Proposition 3.9 we see that
$f^{\prime}(z)\leq f(z)$ and $c^{\prime}(z)\geq c(z)$ for every $z\in T$. ∎

We will decompose this inclusion into a sequence of elementary extensions,
defined as follows.

######  Definition 3.13.

Let $Y=Y(T,x,y,s,t)$ be a crazy diamond.

  * •

An _elementary extension of type $C$_ of $Y$ is a crazy diamond
$Y^{\prime}=Y(T,x^{\prime},y,s+1,t)$, where $x^{\prime}$ is a vertex adjacent
to $x$.

  * •

An _elementary extension of type $F$_ of $Y$ is a crazy diamond
$Y^{\prime}=Y(T,x,y^{\prime},s,t+1)$ where $y^{\prime}$ is a vertex adjacent
to $y$.

  * •

An _elementary extension of type $B_{z}$_ of $Y$ is a crazy diamond
$Y^{\prime}=Y(T^{\prime},x,y,s,t)$ where $T^{\prime}$ is obtained by attaching
an edge at a vertex $z\in T$ such that $l(z)>0$.

In each case, realizing $Y$ as a subset of $T\times\mathbf{R}$ and
$Y^{\prime}$ as a subset of $T^{\prime}\times\mathbf{R}$, we see that there is
a natural inclusion $Y\subset Y^{\prime}$.

Note that in an extension of type $C$ we have

$c^{\prime}(z)=\begin{cases}c(z)&\textrm{ if }d(x,z)<d(x^{\prime},z)\\\
c(z)+1&\textrm{ if not}.\end{cases}$

In the case when $Y$ is degenerate, some of the extensions above are not
possible (since $T$ is too small), but there are also some other possibilities
which appear:

######  Definition 3.14.

  * •

If $Y=Y(\\{x\\},x,x,s,t)$ is vertical, or is a point, an _elementary extension
of type $C_{v}$_ (resp. _of type $F_{v}$_ is the crazy diamond
$Y^{\prime}=Y(\\{x\\},x,x,s+2,t)$ (resp. $Y^{\prime}=Y(\\{x\\},x,x,s,t+2)$

  * •

If $Y=Y(T,x,y,s,t)$ is horizontal, or a point, an _elementary extension of
type $B^{c}$_ (resp. of type $B^{f}$) is of the form
$Y^{\prime}=Y(T^{\prime},x^{\prime},y,s+1,t)$ (resp.
$Y(T^{\prime},x,y^{\prime},s,t+1)$) where $T^{\prime}$ is obtained by
attaching an edge from $x$ to $x^{\prime}$ (resp. $T^{\prime}$ is obtained by
attaching an edge from $y$ to $y^{\prime}$).

Note that an elementary extension of type $C_{v}$ produces a vertical crazy
diamond, while an elementary extension of type $B^{f}$ or $B^{c}$ produces an
horizontal crazy diamond.

######  Lemma 3.15.

Let $Y\subset Y^{\prime}$ be an isometric inclusion of crazy diamonds, and
assume that $Y$ is not degenerate. Then there is a finite sequence
$Y=Y_{0},Y_{1},\dots,Y_{n}=Y^{\prime}$ of crazy diamonds such that
$Y_{i}\subset Y_{i+1}$ is an elementary extension.

###### Proof.

As in Lemma 3.12, we can and shall assume that $T\subset T^{\prime}$. Note
that if $z\in T^{\prime}$ has valency $>1$ then $l^{\prime}(z^{\prime})\geq
1$. Also note that since $c^{\prime}(x)\geq c(x)$, we have $s^{\prime}\geq
s+d(x,x^{\prime})\geq s$.

First assume that $Y$ is non-degenerate. Our sequence of extensions will
proceed as follows. The notations relative to $Y_{i}$ will be the same as for
$Y$ but with index $i$. We use $i$ as a temporary variable and change it after
each step.

By doing two successive extensions of type $C$, we can replace $s$ by $s+2$
without changing $x$. We start by this move, doing $d$ extensions of type $C$,
in order to get that $s_{i}=c_{i}(x)=c^{\prime}(x)$. Note that
$c^{\prime}(x)-c(x)$ must be even since $Y$ is a subcomplex of $Y^{\prime}$.
Then we do in a similar way the right number of extensions of type $F$, in
order to get that $t_{i}=f_{i}(y)=f^{\prime}(y)$. It follows that
$c_{i}(z)=c^{\prime}(z)$ as soon as $d(z,x)<d(z,x^{\prime})$. By doing an
extension of type $C$ again, moving $x$ to the vertex $z_{0}$ adjacent to $x$
and closer to $x^{\prime}$, we get that $c_{i}(z)=c^{\prime}(z)$ for $z$
satisfying $d(z,x)<d(z,x^{\prime})+1$. Repeating the same thing as many times
as necessary, we can get that $c_{i}(z)=c^{\prime}(z)$ for every $z\in T$.
Similarly we can obtain $f_{i}(z)=f^{\prime}(z)$ for every $z\in T$.

If $z\in T$ is of valency $>1$ in $T^{\prime}$, then
$l_{i}(z)=l^{\prime}(z)\geq 1$. Therefore we can perform a extensions of type
$B_{z}$, in order to add as many edges as necessary. Repeating the above
steps, we see that we can get that $T_{i}=T^{\prime}$, $c_{i}=c^{\prime}$ and
$f_{i}=f^{\prime}$, which concludes the argument.

Then let us assume that $Y$ is vertical (but not a point). If $Y^{\prime}$ is
non-degenerate, then there is an alcove in $Y^{\prime}$ which is adjacent to
some panel of $Y$; let $Y_{1}$ be the convex hull of $Y^{\prime}$ and this
alcove. Then $Y\subset Y_{1}$ is an elementary extension of type $B$, and by
the previous argument there exists a finite sequence of elementary extensions
from $Y_{1}$ to $Y^{\prime}$. If $Y^{\prime}$ is also degenerate, then it must
be vertical, then it is just a segment containing $Y$, and one can pass from
$Y$ to $Y^{\prime}$ by adding edges to this segment, that is, by a finite
sequence of elementary extensions of type $C_{v}$ or $F_{v}$.

Now let us assume that $Y$ is horizontal (and not a point). If $Y^{\prime}$ is
non-degenerate, again there is an alcove in $Y^{\prime}$ which is adjacent to
$Y$, and the convex hull $Y_{1}$ of this alcove and $Y$ is an elementary
extension of $Y$ of type either $C$ or $F$. If $Y^{\prime}$ is degenerate,
then it must be horizontal, and then again one can pass from $Y^{\prime}$ to
$Y$ by a sequence of elementary extensions of type $B^{f}$ or $B^{c}$.

Finally, if $Y$ is a point, then there exists a segment in $Y^{\prime}$
containing $Y$, and the inclusion of $Y$ in this segment is an elementary
extensions of type either $C_{v}$, $F_{v}$, $B^{f}$ or $B^{c}$ depending on
the position of this segment in $Y^{\prime}$. In any case applying the
previous argument we can construct a sequence of elementary extensions from
$Y$ to $Y^{\prime}$. ∎

Let $X$ be an $\tilde{A}_{2}$ building of finite thickness $q+1$. Assume that
$i:Y\to Y^{\prime}$ is an isometric embedding of crazy diamonds. Let
$\rho:Y\to X$ be an isometric embedding.

######  Definition 3.16.

An apartment $A\subset X$ is said _compatible with $\rho$_ if it intersects
$\rho(Y)$ and such that

  * •

$\operatorname{pr}(\rho(Y)\cap T)$ is a segment between two leaves of $T$, and

  * •

for every $z\in T$, $\rho(\operatorname{pr}^{-1}(z))\cap A$ is either empty or
the full segment of length $l(z)$.

######  Lemma 3.17.

Any two chambers in $\rho(Y)$ are contained in some compatible apartment.

###### Proof.

Let $\ell$ be a geodesic segment between two leaves of $T$. Then
$\operatorname{pr}^{-1}(\ell)\subset Y$ is isometric to a subset of
$\mathbf{R}^{2}$. Hence by Theorem 2.2, it is contained in an apartment of
$X$, which is therefore a compatible apartment. Now any two edges of $T$ are
contained in some segment between two leaves. Hence any two chambers of $X$
are contained in some maximal rhombus of $Y$, and therefore their image by
$\rho$ is in a compatible apartment. ∎

In view of Lemma 3.15, in order to prove the Proposition, it suffices to prove
it when $Y\subset Y^{\prime}$ is an elementary extension.

######  Lemma 3.18.

Assume that $Y\subset Y^{\prime}$ is an elementary extension of type $F$ or
$C$. Let $\rho:Y\to X$ be an isometric embedding. If $Y$ is non-degenerate,
then there are $q$ possible extensions of $\rho$to
$\rho^{\prime}\in\operatorname{Isom}(Y^{\prime},X)$. If $Y$ is degenerate,
then it is a segment and there are $q+1$ possible extensions.

###### Proof.

We argue for an extension of type $C$, the other one being similar.

Let $e$ be the edge between $x$ and $x^{\prime}$, and let $E$ be the set of
edges of the subtree $T_{x^{\prime}}=\\{z\in T\mid d(z,x^{\prime})<d(z,x)\\}$.
We first remark that $Y^{\prime}$ is obtained from $Y$ by adding one alcove
for each edge in $\\{e\\}\cup E$. For $e^{\prime}\in\\{e\\}\cup E$, let
$d_{e^{\prime}}$ be this alcove, and $c_{e^{\prime}}$ the alcove of $Y$ which
is adjacent to $d_{e^{\prime}}$ (which exists since $\ell(x)>0$, as $Y$ is
non-degenerate), and $\sigma_{e^{\prime}}$ be the panel between
$c_{e^{\prime}}$ and $d_{e^{\prime}}$. Note that in $Y^{\prime}$ the convex
hull of $d_{e}$ and $Y$ is equal to $Y^{\prime}$.

Let $C=\rho(c_{e})$. There are $q$ alcoves in $Y$ which are adjacent to
$\rho(\sigma_{e})$ and distinct from $C$. Let $D$ be such a alcove. We claim
that there exists a unique $\rho^{\prime}:Y^{\prime}\to X$ such that
$\rho^{\prime}(d_{e})=D$.

Indeed, the uniqueness follows from the fact that $X^{\prime}$ is the convex
hull of $d_{e}$ and $Y$. Let us argue for the existence. Let $e^{\prime}\in
E$. By Lemma 3.17 there exists a compatible apartment $A_{0}$ containing $C$
and $\rho(c_{e^{\prime}})$. Furthermore $D\cup(\rho(Y)\cap A_{0})$ is
isometric to a subset of an apartment, hence is contained in an apartment.
Therefore, there exists a compatible apartment $A$ containing $\rho(d_{e})$
and $\rho(c_{e^{\prime}})$. In the apartment $A$, there is a unique alcove
$D_{e^{\prime}}$ which is adjacent to $\rho(\sigma_{e^{\prime}})$ and distinct
from $\rho(c_{e^{\prime}})$.

We define $\rho^{\prime}(d_{e^{\prime}})=D_{e^{\prime}}$. This is well-defined
: indeed, if $A^{\prime}$ is another compatible apartment satisfying the same
conditions, then $A^{\prime}$ contains the convex hull of $\rho(d_{e})$ and
$\rho(c_{e^{\prime}})$, hence also $D_{e^{\prime}}$. Therefore
$D_{e^{\prime}}$ does not depend on the choice of the apartment. Using again
Lemma 3.17, we see that the distance between two chambers can be calculated in
a compatible apartment. It follows that $\rho^{\prime}$ is again isometric.

Therefore, each of the $q$ possible choices of $D$ gives a map
$\rho^{\prime}$, which concludes the proof in the case when $Y$ is non-
degenerate.

If $Y$ is degenerate, then for an extension of type $C$ to be possible, it
must be a horizontal segment in $Y^{\prime}$. In that case, the same argument
applies, but there are in fact possible choices of $D$.

∎

######  Lemma 3.19.

Assume that $Y\subset Y^{\prime}$ is an elementary extension of type $B_{z}$,
where $z$ is a vertex of valency $v$ in $T$. Then there are $q-v+1$ possible
extensions of $\rho:Y\to X$ to $\rho^{\prime}:Y^{\prime}\to X$ if $q-v+1\geq
0$, and $0$ if not.

(Note that if $Y$ is vertical, then the lemma still holds, with $v=0$).

###### Proof.

Let $P$ be the set of panels in $\operatorname{pr}^{-1}(z)\subset Y$. By
assumptions $l(z)>0$, so that $P$ is not empty. Let $l=l(z)$. For $\sigma\in
P$, there is exactly one alcove in $Y^{\prime}$ adjacent to $\sigma$ but which
is not in $Y$. Let $c_{\sigma}$ be this chamber. Note that for every
$\sigma\in P$, we have that $\operatorname{Conv}(c_{\sigma}\cup
Y)=Y^{\prime}$.

Let $\rho:Y\to X$ be an isometric embedding. Let $\sigma\in P$. Let $d$ be an
alcove adjacent to $\rho(\sigma)$ and not contained in $Y$. Then we claim that
there exists exactly one map $\rho^{\prime}:Y^{\prime}\to X$ such that
$\rho^{\prime}(c_{\sigma})=d$. The result follows immediately since there are
$q+1$ alcoves in $Y$ adjacent to $\sigma$ and $v$ of them are contained in
$\rho(Y)$.

So let us prove the claim. The uniqueness is clear since
$Y^{\prime}=\operatorname{Conv}(c_{\sigma}\cup Y)$. The existence follows from
a similar argument as above. Let $e$ be the new edge in $Y^{\prime}$, and let
$A$ be a compatible apartment for $\rho$ containing $\rho(\sigma)$ (and
therefore $\rho(\operatorname{pr}^{-1}(z))$). The wall of $A$ containing
$\rho(\sigma)$ divides $A$ into two half-spaces $\alpha$ and $\beta$. Then
$\alpha\cup d$ is contained in some apartment $A_{1}$. In $A_{1}$ we see that
the convex hull of $\rho(Y)\cap A_{1}$ and $d$ is formed by adding the $2l+1$
alcoves not in $\alpha$ and sharing either an edge or a vertex with some panel
$\rho(\sigma^{\prime})$, for some $\sigma^{\prime}\in P$.

Note that this description is independent on the choice of $A$ or $\alpha$
(indeed, it is just given by the convex hull of $d$ and all panels
$\rho(\sigma^{\prime})$, for $\sigma^{\prime}\in P$). Since every alcove of
$\rho(Y)$ is contained in a compatible apartment containing also
$\rho(\sigma)$, it follows that the convex hull of $\rho(Y)$ and $d$ is formed
by adding exactly these $2l+1$ alcove. So we can extend $\rho^{\prime}$ in the
obvious way to the $2l+1$ alcoves that are in $Y^{\prime}\setminus Y$. Since
any alcove in $\rho(Y)$ is contained in a compatible apartment containing
$\rho(\operatorname{pr}^{-1}(z))$, we see that the extension $\rho^{\prime}$
is indeed isometric. ∎

######  Lemma 3.20.

Let $Y\subset Y^{\prime}$ be an elementary extension of type $F_{v}$ or
$C_{v}$ or $B^{c}$ or $B^{f}$, and let $\rho\in\operatorname{Isom}(Y,X)$ If
$Y$ is not a point, then there are $q^{2}$ possible extensions of $\rho$ to
$\rho^{\prime}\in\operatorname{Isom}(Y^{\prime},X)$. If $Y$ is a point, then
there are $q^{2}+q+1$ such extensions.

###### Proof.

If $Y$ is a point, then $Y\subset Y^{\prime}$ is just the inclusion of a point
in an edge, and this is just counting the degree of vertices in the 1-skeleton
of $X$.

If $Y$ is a segment, then any of these elementary extensions amount to adding
an edge at an extremal vertex. In the link of this vertex, this edge must be
opposite the image of $Y$. There are $q^{2}$ such edges. ∎

###### Proof of Theorem 3.1.

Let $Y,Y^{\prime}$ be crazy diamonds and $i:Y\to Y^{\prime}$ be an isometry.
Let $\rho:Y\to X$ be an isometry. By Lemma 3.15, there exists a sequence of
crazy diamonds $Y=Y_{0},Y_{1},\dots,Y_{n}=Y$, such that $i_{k}:Y_{k}\subset
Y_{k+1}$ is an elementary extension. The number of maps
$\rho^{\prime}:Y^{\prime}\to X$ such that $\rho^{\prime}\circ i=\rho$ is the
product over $k$ of the numbers of maps $\rho_{k+1}$ such that
$\rho_{k+1}\circ i_{k}=\rho_{k}$. By Lemma 3.18, Lemma 3.19 and 3.20, this
number only depends on the inclusion $Y_{i}\subset Y_{i+1}$. Therefore the
desired number only depends on the inclusion $Y\subset Y^{\prime}$. ∎

###  3.2 Flows and measures

In this section, we revisit various constructions of measures and dynamical
systems associated to buildings, in the light of the construction of
prouniform measures.

######  Convention.

In this text, the index of the measure will be an indication of the space on
which this measure is defined. The exponent will usually denote some origin or
restriction.

We also will use the letter $\mu$ only for (restricted or not) prouniform
measures.

####  3.2.1 Harmonic measures and the Cartan Flow

Let us start again with $X$ an $\tilde{A}_{2}$ building, and $\Sigma$ a model
$\tilde{A}_{2}$ apartment. Recall that the property $P_{\Upsilon}$ was defined
to be "being a finite convex subset of the wall-tree $\Upsilon$". Note that
every convex subset of $\Upsilon$ is ind-$P_{\Upsilon}$ (as the convex hull of
a finite set in $\Upsilon$ is finite).

Recall that we have fixed an origin $0\in\Sigma$, and a sector
$\Lambda\subset\Sigma$ based at $0$. Note that $\Sigma$ and $\Lambda$ are
ind-$P_{\Upsilon}$. Let $o\in X$. The set $\Delta$ of Weyl chambers in the
boundary of $X$ can be identified with the set of sectors based at $o$.
Therefore we can define the restricted prouniform measure $\mu_{\Lambda}^{o}$
on $\operatorname{Isom}(\Lambda,X)^{o}\simeq\Delta$. In the following of the
text we will denote this measure $\mu_{\Delta}^{o}$. This measures has been
studied for a long time, previous definitions can be found in [Parkinson] or
more recently [RemyTrojan, Proposition 6.1].

Recall that we have a natural map from
$\operatorname{pr}_{+}:\Delta\to\Delta_{+}$, associating to a Weyl chamber its
vertex of type $+$. This map is $\operatorname{Aut}(X)$-equivariant and
continuous. Similarly we have $\operatorname{pr}_{-}:\Delta\to\Delta_{-}$. We
also define the pushforward measures
$\mu_{+}^{o}=(\operatorname{pr}_{+})_{*}\mu_{\Delta}^{o}\in\operatorname{Prob}(\Delta_{+})$
and
$\mu_{-}^{o}=(\operatorname{pr}_{-})_{*}\mu_{\Delta}^{o}\in\operatorname{Prob}(\Delta_{-})$.

######  Remark 3.21.

The measures $\mu_{+}^{o}$ and $\mu_{-}^{o}$ are not prouniform measures per
se. However, one could define a slightly different notion a prouniform
measures, taking into account the type and considering only type-preserving
simplicial isometries. In this sense, identifying $\Delta_{+}$ with the set of
rays of type $+$ starting from $o$, one could check that we would indeed get
the measure $\mu_{+}^{o}$. This point of view should be very important if one
tries to extend the constructions we make here to the setting of building of
other types, e.g $\tilde{C}_{2}$ and $\tilde{G}_{2}$. We decided not to
elaborate on this further in this paper, in order to keep things simpler.

The measure $\mu_{\Delta}^{o}$ depends of course on the choice of the point
$o$. However, its class does not. More precisely, one can calculate the Radon-
Nikodym derivative as follows. Let $x,y\in X$ be two vertices. For
$\omega\in\Omega$, the sectors $Q(x,\omega)$ and $Q(y,\omega)$ intersect, by
definition, so let $z\in Q(x,\omega)\cap Q(y,\omega)$. Let
$\lambda=\sigma(x,z)$ and $\lambda^{\prime}=\sigma(y,z)$. Then we define the
_horofunction_ associated to $\omega$ by
$h_{\omega}(x,y)=\lambda-\lambda^{\prime}$. It is easily seen that this
quantity does not depend on the choice of $z$.

Also recall that, for $\lambda\in\Lambda$, $\ell(\lambda)$ denotes the length
of $\lambda$. The following proposition is proven for example in [Parkinson,
Theorem 3.17].

######  Proposition 3.22.

For every vertices $x,y\in X$,

  1. 1.

the measures $\mu_{\Delta}^{x}$ and $\mu_{\Delta}^{y}$ are absolutely
continuous with respect to each other

  2. 2.

for every $\omega\in\Delta$,

$\frac{d\mu_{\Delta}^{x}}{d\mu_{\Delta}^{y}}(\omega)=q^{2\ell(h_{\omega}(x,y))}$

The set $\Delta\times\Delta$ contains a dense open subset
$\Delta_{\mathrm{op}}$ formed by the set of pairs of chambers which are
opposite. This set has full $\mu^{o}_{\Delta}\times\mu^{o}_{\Delta}$-measure
(see [BCL, Proposition 6.8]). In [BCL, §6.2] we constructed a
$\operatorname{Aut}(X)$-invariant measure on $\Delta\times\Delta$, denoted $m$
there (and that we will denote $m_{\mathrm{op}}$ here). This measure was given
by the following formula:

$dm(C,C^{\prime})=q^{-2\ell(\beta_{o}(C,C^{\prime}))}d\mu_{\Delta}^{o}(C)d\mu_{\Delta}^{o}(C^{\prime})$

(where $\beta_{o}(C,C^{\prime})$ is some kind of Gromov product on
$\Delta_{\mathrm{op}}$, which satisfies in particular that
$\beta_{o}(C,C^{\prime})=0$ if and only if $o$ is in the apartment containing
$C$ and $C^{\prime}$). Using [BCL, Lemma 6.9] one proves that this formula
does not depend on the choice of the origin $o$, and that it gives indeed an
$\operatorname{Aut}(X)$-invariant Radon measure. Similarly in [BCL, Definition
6.14] we defined a measure $m_{\pm}$ on the set
$\Delta_{\pm,\mathrm{op}}\subset\Delta_{-}\times\Delta_{+}$ of opposite
vertices, which is in the class of $\mu_{-}^{o}\times\mu_{+}^{o}$. The natural
map
$\operatorname{pr}_{-}\times\operatorname{pr}_{+}:\Delta_{\mathrm{op}}\to\Delta_{\pm,\mathrm{op}}$
is measure-preserving.

We define the _Cartan Flow_ as $\mathscr{F}=\operatorname{Isom}(\Sigma,X)$,
equipped with the prouniform measure, denoted $\mu_{\mathscr{F}}$. This is
again a measure which has been considered before, although maybe it is not so
well-known (see [Pansu] for a previous construction). The space $\mathscr{F}$
is equipped with commuting actions of $\operatorname{Aut}(\Sigma)$ (which is
virtually $\mathbb{Z}^{2}$) and $\operatorname{Aut}(X)$, which are measure-
preserving.

Note however that in [BCL] we considered a slightly different construction of
the measure. Indeed, identifying $\mathscr{F}$ with
$\Delta_{\mathrm{op}}\times\Sigma$, we equipped $\mathscr{F}$ with the product
measure $\zeta_{\mathscr{F}}:=m_{\mathrm{op}}\times\lambda$, where $\lambda$
is the counting measure on $\Sigma$.

In order to apply the results of [BCL] we are led to check the following:

######  Lemma 3.23.

The measure $\zeta_{\mathscr{F}}$ on $\mathscr{F}$ defined (and denoted
$\zeta^{\prime}$) in [BCL, Theorem 6.21] is equal to the prouniform measure
$\mu_{\mathscr{F}}$.

###### Proof.

Write $\mathscr{F}$ as a disjoint union of the sets $\mathscr{F}^{x}$ for
$x\in X$, where $\mathscr{F}^{x}$ is the set of embeddings $\phi:\Sigma\to X$
such that $\phi(0)=x$. All the sets $\mathscr{F}^{x}$ are compact subsets of
$\mathscr{F}$, and by construction of $\mu_{\mathscr{F}}$ we have
$\mu_{\mathscr{F}}(\mathscr{F}^{x})=1$ for every $x$. Using the invariance of
$\zeta_{\mathscr{F}}$ by translations we also see that these sets have all the
same $\zeta_{\mathscr{F}}$ measure, which is therefore positive.

Therefore, it remains to see that $\mu_{\mathscr{F}}=\zeta_{\mathscr{F}}$ in
restriction to $\mathscr{F}^{x}$. To understand these measures, first note
that there is a homeomorphism
$\Psi:\mathscr{F}^{x}\to\Delta^{x}_{\mathrm{op}}\subset\Delta^{2}$ from
$\mathscr{F}^{x}$ to the set $\Delta^{x}_{\mathrm{op}}$ formed by pairs of
Weyl chamber $(C,C^{\prime})$ with $C$ opposite $C^{\prime}$ and $x$ in an
apartment containing $C$ and $C^{\prime}$. For
$(C,C^{\prime})\in\Delta^{x}_{\mathrm{op}}$ we have
$\beta_{x}(C,C^{\prime})=0$, so by construction, the pushforward of
$\zeta_{\mathscr{F}}$ by $\Psi$ is the restriction of the measure
$\mu^{x}_{\Delta}\times\mu^{x}_{\Delta}$ to $\mathscr{F}^{x}$.

Recall that $\Lambda\subset\Sigma$ is our choice of a sector based at $0$. For
$y$ in the interior of $\Lambda$, denote
$\Sigma^{y}=\operatorname{Conv}(y,-y)$. Note that $\Sigma$ is the union of all
the finite convex subsets $\Sigma^{y}$. For an embedding
$\phi^{y}:\Sigma^{y}\to X$ such that $\phi^{y}(0)=x$, denote
$\mathscr{F}^{\phi^{y}}=\operatorname{Isom}(\Sigma,X)^{\phi^{y}}$ the subset
of $\mathscr{F}$ formed by all $\phi:\Sigma\to X$ such that
$\phi_{|\Sigma^{y}}=\phi^{y}$. By construction,
$\mu_{\mathscr{F}}(\mathscr{F}^{\phi^{y}})$ depends only on $y$ and not on the
choice of $\phi^{y}$. Since $\Sigma$ is the union of all $\Sigma^{y}$, this
characterizes $\mu_{\mathscr{F}}^{x}$, so that it suffices to prove that
$(\zeta_{\mathscr{F}})_{|\mathscr{F}^{x}}$ satisfies the same property.

Now note that defining $\phi^{y}$ amounts to choosing $y_{1}:=\phi^{y}(y)$ and
$y_{2}:=\phi^{y}(-y)$. Hence $\Psi(\mathscr{F}^{\phi^{y}})$ is the set of
pairs of Weyl chambers $(C,C^{\prime})$ in $\Delta^{x}_{\mathrm{op}}$ such
that $C\in\Omega_{x}(y_{1})$ and $C^{\prime}\in\Omega_{x}(y_{2})$. Conversely,
if $C\in\Omega_{x}(y_{1})$ and $C^{\prime}\in\Omega_{x}(y_{2})$, then using
Theorem 2.2 one can check that there exists an apartment containing $x,C$ and
$C^{\prime}$, hence $(C,C^{\prime})\in\Delta_{\mathrm{op}}^{x}$. Therefore we
have $\Psi(\mathscr{F}^{\phi^{y}})=\Omega_{x}(y_{1})\times\Omega_{x}(y_{2})$,
hence
$\Psi_{*}\zeta_{\mathscr{F}}(\mathscr{F}^{\phi^{y}})=\mu^{x}_{\Delta}(\Omega_{x}(y_{1}))\times\mu_{\Delta}^{x}(\Omega_{x}(y_{2}))$,
which only depends on $\sigma(x,y)$. This concludes the proof. ∎

####  3.2.2 Martingales

Our goal in this section is to reinterpret the Martingale Convergence Theorem
(stated in Theorem 1.15) and draw consequences in our particular setting.

Fix $o\in X$. For $v\in\Delta_{+}$, let
$\mu_{v}^{o}\in\operatorname{Prob}(\Delta)$ be the restricted prouniform
measure on $\operatorname{Isom}(\Lambda,X)$ which sends the boundary wall of
type $+$ of $\Lambda$ to the ray $[ov)$. Clearly $\mu_{v}^{o}$ is supported on
$\operatorname{Res}(v)$. Furthermore by Lemma 1.10 the measure $\mu_{v}^{o}$
for $v\in\Delta_{+}$ are also obtained as a disintegration of
$\mu_{\Delta}^{o}$ along a fiber of the map $\Delta\to\Delta_{+}$.

Let $\lambda\in\Lambda$. Consider the equivalence relation $\sim_{\lambda}$ on
$\operatorname{Ch}(\Delta)$, defined as follows : $C\sim_{\lambda}C^{\prime}$
if and only if $Q(o,C)\cap Q(o,C^{\prime})$ contains a point in
$V_{\lambda}(o)$. Let $\mathcal{F}_{\lambda}$ be the $\sigma$-algebra
consisting of Borel sets which are $\sim_{\lambda}$-invariant. Note that if
$\lambda^{\prime}\geq\lambda$ then $\sim_{\lambda^{\prime}}$ is a refinement
of $\sim_{\lambda}$, and therefore
$\mathcal{F}_{\lambda^{\prime}}\subset\mathcal{F}_{\lambda}$. For
$C\in\Delta$, let $\lambda(C)$ be the point in $V_{\lambda}(o)\cap Q(o,C)$,
and write $\Omega_{\lambda}(C)=\Omega_{o}(\lambda(C))$. If $\lambda$ is on the
wall of type $+$ of $\Lambda$, then $\lambda(C)$ only depends on
$v=\operatorname{pr}_{+}(C)$, and we can also write
$\Omega_{\lambda}(C)=\Omega_{\lambda}(v)$. By definition we have, for $f\in
L^{\infty}(\Delta)$,

$\mathbf{E}(f\mid\mathcal{F}_{\lambda})(C)=\frac{1}{\mu_{\Delta}^{o}(\Omega_{\lambda}(C))}\int_{\Omega_{\lambda}(C)}f(x)d\mu_{\Delta}^{o}(x)$

Let $\mathcal{F}_{n}$ be the subalgebra $\mathcal{F}_{\lambda_{n}}$ where
$\lambda_{n}$ is the point on the wall of type $+$ of $\Lambda$ at distance
$n$ from $0$. It is clear that $\mathcal{F}_{n+1}\supset\mathcal{F}_{n}$. Let
$\mathcal{F}_{\infty}$ be the algebra generated by all the $\mathcal{F}_{n}$.
By construction it is also the algebra obtained as a pullback from the factor
$\Delta\to\Delta_{+}$. Therefore, we get for every
$C\in\operatorname{Ch}(\Delta)$, denoting $v=\operatorname{pr}_{+}(C)$, that
for $f\in L^{\infty}(\Delta)$

$\mathbf{E}(f\mid\mathcal{F}_{\infty})(C)=\int_{\operatorname{Res}(v)}f(\xi)d\mu_{v}^{o}(C)$

For $v\in\Delta_{+}$, denote $\lambda_{n}(v)$ the $n$-th vertex on the ray
$[ov)$ and note that
$\Omega_{\lambda_{n+1}}(v)\subset\Omega_{\lambda_{n}}(v)$. Let
$E_{n}(v)=\Omega_{\lambda_{n}}(v)\setminus\Omega_{\lambda_{n+1}}(v)$.

More generally, for $m\geq 0$, let $W_{m}$ be the half-wall of $\Lambda$
parallel to the wall of type $+$ and starting at distance $m$ from $0$. Let
$\lambda_{n,m}$ be the vertex of $W_{m}$ at distance $n$ from the origin of
$W_{m}$.

Let $\mathscr{F}_{m,\infty}$ be the $\sigma$-algebra generated by all the
$\\{\mathscr{F}_{\lambda_{m,n}}\\}_{n\in\mathbf{N}}$. It is generated by the
sets that we will denote $\Omega_{m,\infty}(\xi)$ which are defined as
$\Omega_{m,\infty}(\xi)=\bigcap_{n\in\mathbf{N}}\Omega_{\lambda_{m,n}}(\xi)$.
Note that $\Omega_{m,\infty}(\xi)$ can also be seen as the set of all
$\xi^{\prime}$ such that $Q(o,\xi^{\prime})$ contains the "strip"
$\operatorname{Conv}([o,v),\\{\lambda_{m,n}(\xi)\\}_{n\in\mathbf{N}})=\operatorname{Conv}([o,v),\lambda_{m,0}(\xi))$
(where $v=\operatorname{pr}_{+}(\xi)$, see Figure 1). In particular
$\Omega_{m,\infty}(\xi)\subset\operatorname{Res}(v)$ , and
$\mu_{v}^{o}(\Omega_{m,\infty}(\xi))>0$.

Figure 1: The convex hull of the $\lambda_{m,n}$, $n\geq 0$

In that setting, Theorem 1.15 reads as follows:

######  Lemma 3.24.

Let $f\in L^{\infty}(\Delta)$. For almost every $\xi\in\Delta$ we have

$\lim_{n\to+\infty}\frac{1}{\mu_{\Delta}^{o}(\Omega_{\lambda_{m,n}}(\xi))}\int_{\Omega_{\lambda_{m,n}}(\xi)}f(\eta)d\mu_{\Delta}^{o}(\eta)=\frac{1}{\mu_{v}^{o}(\Omega_{m,\infty)}(\xi)}\int_{\Omega_{m,\infty}(\xi)}f(\eta)d\mu_{v}^{o}(\eta)$

For $\xi\in\Delta$, let
$E_{m,n}(\xi)=\Omega_{\lambda_{m,n}}(\xi))\setminus\Omega_{\lambda_{m,n+1}}(\xi))$.
Note that
$\mu_{\Delta}^{o}(\Omega_{\lambda_{m,n+1}}(\xi))=\frac{1}{q^{2}}\mu_{\Delta}^{o}(\Omega_{\lambda_{m,n}}(\xi)))$.
Indeed, if $\eta$ is such that $\lambda_{m,n}(\xi)\in Q(o,\eta)$ then there
are $q^{2}$ possibilities for the choice of $\lambda_{m,n+1}(\eta)$.

######  Lemma 3.25.

Let $f\in L^{\infty}(\Delta)$. For every $m\in\mathbf{N}$ and
$\mu_{\Delta}^{o}$-almost every $\xi$, denoting $v=\operatorname{pr}_{+}(\xi$)
we have

$\lim_{n\to+\infty}\frac{1}{\mu_{\Delta}^{o}(E_{m,n}(\xi))}\int_{E_{m,n}(\xi)}f(\eta)d\mu_{\Delta}^{o}(\eta)=\frac{1}{\mu_{u}^{o}(\Omega_{m,\infty}(\xi))}\int_{\Omega_{m,\infty}(\xi)}f(\eta)d\mu_{v}^{o}(\eta)$

In particular, for almost $\mu_{+}^{o}$-every $v$ we have

$\lim_{n\to+\infty}\frac{1}{\mu_{\Delta}^{o}(E_{n}(v))}\int_{E_{n}(v)}f(C)d\mu_{\Delta}^{o}(C)=\int_{\operatorname{Res}(v)}f(C)d\mu^{o}_{v}(C)$

###### Proof.

The second statement follows from the first by taking $m=0$ (using also the
fact that $\operatorname{pr}:\Delta\to\Delta_{+}$ is measure-preserving).

So let us prove the first statement. Fix $\xi$ such that Lemma 3.24 holds.
Write $\Omega_{m,n}=\Omega_{\lambda_{m,n}}(\xi)$ and $E_{m,n}=E_{m,n}(\xi)$.
By the above estimate we have
$\mu(E_{m,n})=(1-\frac{1}{q^{2}})\mu(\Omega_{m,n})$.

It follows that

$\displaystyle\frac{1}{\mu_{\Delta}^{o}(E_{m,n})}\int_{E_{m,n}}f(C)d\mu_{\Delta}^{o}(C)$
$\displaystyle=\frac{q^{2}}{q^{2}-1}\cdot\frac{1}{\mu_{\Delta}^{o}(\Omega_{m,n})}\int_{E_{n}}f(C)d\mu_{\Delta}^{o}(C)$
$\displaystyle=\frac{q^{2}}{q^{2}-1}\cdot\frac{1}{\mu_{\Delta}^{o}(\Omega_{m,n})}\left(\int_{\Omega_{m,n}}f(C)d\mu_{\Delta}^{o}(C)-\int_{\Omega_{m,n+1}}f(C)d\mu_{\Delta}^{o}(C)\right)$
$\displaystyle=\frac{q^{2}}{q^{2}-1}\cdot\frac{1}{\mu_{\Delta}^{o}(\Omega_{m,n})}\int_{\Omega_{m,n}}f(C)d\mu_{\Delta}^{o}(C)$
$\displaystyle\qquad\qquad-\frac{1}{q^{2}-1}\cdot\frac{1}{\mu_{\Delta}^{o}(\Omega_{m,n+1})}\int_{\Omega_{m,n+1}}f(C)d\mu_{\Delta}^{o}(C)$

which converges by assumption to

$(\frac{q^{2}}{q^{2}-1}-\frac{1}{q^{2}-1})\frac{1}{\mu_{v}^{o}(\Omega_{m,\infty}(\xi))}\int_{\Omega_{m,\infty}(\xi)}f(\eta)d\mu_{u}^{o}(\eta)=\frac{1}{\mu_{u}^{o}(\Omega_{m,\infty}(\xi))}\int_{\Omega_{m,\infty}(\xi)}f(\eta)d\mu_{v}^{o}(\eta).$

∎

Finally, let us record the following remark.

######  Lemma 3.26.

Let $\xi\in\Delta$ and $v=\operatorname{pr}_{+}(\xi)$. We have
$\Omega_{m,n}(\xi)\cap E_{n}(v)=E_{m,n}(\xi)$.

###### Proof.

If $x\in\Omega_{m,n+1}(\xi)$ then
$\operatorname{Conv}(o,\lambda_{m,n+1}(\xi))\subset Q(o,x)$ and therefore
$\lambda_{n+1}(v)\in Q(o,x)$, so that $x\not\in E_{n}$. Hence we have the
inclusion $\Omega_{m,n}(\xi)\cap E_{n}(v)\subset E_{m,n}(\xi)$.

Conversely, if $x\in E_{m,n}(\xi)$ then
$x\in\Omega_{m,n}(\xi)\supset\Omega_{n}(u)$ by definition, and as
$\lambda_{m,n+1}(\xi)\in\operatorname{Conv}(\lambda_{n+1}(v),\lambda_{m,n}(\xi))$
we have $x\not\in\Omega_{n+1}(v)$. ∎

####  3.2.3 Singular flow

Recall our definition of the model wall-tree $\Upsilon$, which is isometric to
$T\times\mathbf{R}$ (where $T$ is a $(q+1)$-regular tree). We also define the
space of _marked wall trees_ as
$\tilde{\mathscr{W}}=\operatorname{Isom}(\Upsilon,X)$. Note that
$\operatorname{Aut}(\Upsilon)$ acts on $\tilde{\mathscr{W}}$ (by
precomposition), and $\operatorname{Aut}(\Upsilon)$ contains the subgroup
$S_{\Upsilon}$ isomorphic to $\mathbf{Z}$ (acting by translation on the factor
$\mathbf{R}$ of the decomposition $T\times\mathbf{R}$). Furthermore, we have a
canonical projection
$p_{T}:\operatorname{Aut}(\Upsilon)\to\operatorname{Aut}(T)$, which is
surjective.

Unfortunately, it turns out that the prouniform measure on this space is not
adapted to what we can do ; in fact we need to consider a smaller subset. The
_singular flow_ $\mathscr{W}$ is a subset of $\tilde{\mathscr{W}}$, which is
defined as one (arbitrary) equivalence class of $\tilde{\mathscr{W}}$, for an
equivalence relation defined in [BCL, Definition 5.17]. In particular, by
construction, the space $\mathscr{W}$ is

  * •

Invariant by the action of $S_{\Upsilon}$,

  * •

Invariant by the action of $p_{T}^{-1}(\Pi)$,

  * •

Invariant by $\Gamma$.

(Recall that $\Pi$ is the closure of the projectivity group, as indicated in
the list of notations.)

Note that there is a natural map $\pi:\mathscr{W}\to\Delta_{\pm,\mathrm{op}}$,
associating to an embedding the image of the two endpoints of the $\mathbf{R}$
factor. By construction, for every $(u,v)\in\Delta_{{\pm,\mathrm{op}}}$ the
fiber $\pi^{-1}(u,v)$ is in bijection with a closed group
$\tilde{\Pi}<\operatorname{Aut}(T)$ which contains $\Pi$.

We defined in [BCL, Theorem 6.21] a measure $\zeta$ on $\mathscr{W}$, which we
will now denote $\zeta_{\mathscr{W}}$. This measure is such that:

  * •

The map $\pi$ is measure-class preserving

  * •

The disintegration of $\zeta$ along the map $\pi$ is the Haar measure on
$\tilde{\Pi}$.

More generally, the measure $\zeta_{\mathscr{W}}$ is also such that all the
natural maps given in the commutative diagram [BCL, 5.4 (3)] are measure-class
preserving.

######  Remark 3.27.

Let us emphasize that the measure $\zeta_{\mathscr{W}}$ is _not_ in general
the prouniform measure on $\operatorname{Isom}(\Upsilon,X)$ ! The reason we
need a different measure is because the measure $\zeta_{\mathscr{W}}$ will be
ergodic under the action of $\Gamma\times S_{\Upsilon}$, while the measure
$\mu_{\mathscr{W}}$ may not. This will be crucial in our proof of Proposition
4.8.

##  4 Approximation of the singular flow

In this section, we use the singular flow $\mathscr{W}$ defined in Section 3.2
to construct sequences of singular hyperbolic elements, with a prescribed
transverse action. We then prove that this sequence of elements have a nice
dynamical action on the boundary $\Delta$.

###  4.1 The argument for the tree

Before diving into the details of the proof, we will start by describing the
full argument in an easier case, namely, the case of a tree. Many of the proof
are similar to the case of the building, so we will not give full details, as
our main purpose here is to illustrate the ideas. Later, when we will deal the
building case we will be more cautious with the details.

Let $T$ be a locally finite regular tree and $\partial T$ its boundary. Let
$\Gamma$ be a group acting properly and cocompactly on $T$. We endow $\partial
T$ with the restricted prouniform measure $\mu_{\partial T}^{o}$ on
$\operatorname{Isom}([0,+\infty),T)$ (where $o\in T$ is an arbitrary origin).

The main goal in this section is the proof of the following.

######  Proposition 4.1.

For every $f\in L^{\infty}(\partial T,\mu_{\partial T}^{o})$, and almost every
$\xi\in\partial T$, there exists a sequence
$(\gamma_{n})\in\Gamma^{\mathbf{N}}$ such that $\gamma_{n}\cdot f$ converges
to (the constant function) $f(\xi)$ in weak-* topology.

####  4.1.1 A recurrence argument

Consider the geodesic flow $\mathscr{G}$ on the tree $T$: it is the set of
parametrized geodesics on $T$, equipped with the prouniform measure
$\mu_{\mathscr{G}}$ and the action of the shift $S_{\mathscr{G}}$. It is
classical, and not hard to see, that the action of $\Gamma$ on
$\mathscr{G}/S_{\mathscr{G}}$ is ergodic, or equivalently that the action of
$S_{\mathscr{G}}$ on $\Gamma\backslash\mathscr{G}$ is ergodic.

The first step of our proof is to find contracting sequence in every direction
in $\Gamma$. Note that since $\mu_{\partial T}^{o}$ is non-atomic,
$(\mu_{\partial T}^{o})^{2}$-almost surely two points $(\eta,\xi)\in\partial
T\times\partial T$ are distinct, hence linked by a geodesic (denoted
$(\eta,\xi)$).

Say that a sequence $(\gamma_{n})$ of automorphisms of $T$ is
$(\eta,\xi)$-contracting if for every finite segment $I$ contained in the
geodesic $(\eta,\xi)$, for $n$ large enough $\gamma_{n}$ acts on $I$ by
translating it towards $\xi$ by a length at least $n$.

######  Lemma 4.2.

For $(\mu_{\partial T}^{o})^{2}$-almost every $(\eta,\xi)\in\partial
T\times\partial T$ there exists a $(\eta,\xi$)-contracting sequence
$(\gamma_{n})_{n\in\mathbf{N}}$ of elements of $\Gamma$.

The analogous statement of the building is proved in Proposition 4.5 below.
The argument is roughly the same in both cases, except that Proposition 4.5
uses the singular flow on the building instead of the geodesic flow on the
tree.

###### Proof.

Let $I$ be a finite segment. Then the shift $S_{\mathscr{G}}$ on
$\Gamma\backslash\mathscr{G}$ preserves (by cocompactness of $\Gamma$) a
probability measure in the class of the pushforward of $\mu_{\mathscr{G}}$.
Hence it is recurrent. If $I$ is a finite segment, then the set of geodesics
containing $I$ has positive measure, thus is $S_{\mathscr{G}}$-recurrent,
which means that there is a sequence $\gamma_{n}$ such that
$\gamma_{n}S_{\mathscr{G}}^{-N}$ fixes $I$ (for some $N>n$).

This almost gives the sequence $\gamma_{n}$ that we want, except that we do
not have any control over the endpoints $\xi$ and $\eta$. To get the Lemma for
almost every $\xi$ and $\eta$ one needs to use the ergodicity of
$S_{\mathscr{G}}$. ∎

Note that for such a sequence $(\gamma_{n})$ we have that
$\gamma_{n}\xi^{\prime}$ converges to $\xi$, for every point
$\xi^{\prime}\neq\eta$. In particular if $f$ is continuous on $\partial T$
then $\gamma_{n}f$ converges to $f(\xi)$ pointwise. Of course it is far from
sufficient to get Proposition 4.1 as we only have a measurable function which
is only almost everywhere defined.

####  4.1.2 Martingales and Lebesgue differentiation

The second step of our argument is to apply some sort of disintegration. This
is the analogous of what is done in §3.2.2 above.

For $o\in T$ fixed, and $x\in T$ denote $\Omega(x)\subset\partial T$ the
shadow of $x$ seen from $o$, that is, the set of endpoints $\xi\in\partial T$
for which $x$ is on the geodesic ray $[o\xi)$. For $n\in\mathbf{N}$,
$\xi\in\partial T$, denote $\Omega_{n}(\xi)$ the shadow of the $n$th vertex of
$[o\xi)$.

The sets $\Omega_{n}(\xi)$ can be seen to be balls around $\xi$, for a natural
metric on $\partial T$. Hence the following is a direct application of the
Lebesgue Differentiation Theorem. In that case, one can also easily apply the
Martingale Convergence Theorem to get the same result.

######  Lemma 4.3.

For every $f\in L^{\infty}(\partial T,\mu_{\partial T}^{o})$, for
$\mu_{\partial T}^{o}$-almost every point $\xi$ we have

$\lim_{n}\frac{1}{\mu_{\partial
T}^{o}(\Omega_{n}(\xi))}\int_{\Omega_{n}(\xi)}fd\mu_{\partial T}^{o}=f(\xi)$

In fact, using a classical trick (compare for example with [Folland, p.98])
one can prove the same convergence when averaging over a subset of
$\Omega_{n}(\xi)$ which has a measure equal to a fixed (independent of $n$)
proportion of $\Omega_{n}(\xi)$ (for the measure $\mu_{\partial T}^{o}$).
Furthermore one can easily check that the "sphere"
$E_{n}(\xi):=\Omega_{n}(\xi)\setminus\Omega_{n+1}(\xi)$ has positive measure,
and in fact takes a fixed proportion of the ball $\Omega_{n}(\xi)$. Hence the
Lemma also holds when replacing the average of $f$ over $\Omega_{n}(\xi)$ with
its average over $E_{n}(\xi)$.

An important difficulty (and also a crucial point of the proof) in the
building case is that the analogous limit that we get will not be a constant
anymore, but a function on a smaller set.

####  4.1.3 The detecting flow

The final step in the proof of Proposition 4.1 is to relate the sequence
$(\gamma_{n})$ to the averages over $E_{n}(\xi)$. While this can also be done
directly in the case of the tree, we found it easier in the building case to
use yet another space, which we call the _detecting flow_.

We define a space $M$, which we think of as a model geodesic endowed with
antennas, or a _comb_ , by setting

$M=\left(\mathbf{R}\times\\{0\\}\right)\cup\left(\mathbf{Z}\times[0,\infty)\right)\subset\mathbf{R}^{2},$

which we view as the union of a geodesic ($\mathbf{R}\times\\{0\\}$) with
antennas at each vertex (the lines $\\{n\\}\times[0,\infty)$) (see Figure 2
below). We define the detecting geodesic flow on $T$ to be

$\mathcal{J}=\mathrm{Isom}(M,T).$

Figure 2: The model space for the detecting flow on a tree

We endow $\mathcal{J}$ with the prouniform measure $\mu_{\mathcal{J}}$. For
every $n\in\mathbf{Z}\cup\\{\pm\infty\\}$ we define a map
$\pi_{n}:\mathcal{J}\to\partial T$ as the limit of the image of the ray
$\\{n\\}\times[0,\infty)$ (these are the "antenna" maps). On the other hand,
we can also define $\pi_{\infty},\pi_{-\infty}:\mathcal{I}\to\partial T$ as
the limit of the geodesic rays $[0,+\infty)\times\\{0\\}$ and $(-\infty,0]$.

We define the "shift" on $\mathbf{R}^{2}$ by $S(x,y)=(x+1,y)$ and note it
leaves $M$ invariant, hence it induces an automorphism of $\mathcal{J}$, which
commutes with the action of $\Gamma$. Furthermore we have easily that
$\pi_{n}\circ S=\pi_{n+1}$.

The maps $\pi_{n}$ (for $n\in\mathbf{Z}\cup\\{\pm\infty\\}$) define pullbacks
$\pi_{n}^{*}:L^{\infty}(\partial T)\to L^{\infty}(\mathcal{J})$. (Note that
$\pi_{n}:\mathcal{J}\to\partial T$ is measure-class preserving). However,
since our favorite contracting sequence $(\gamma_{n})$ depends on a choice of
points $\xi$ and $\eta$, we need to first proceed to a disintegration and fix
the two endpoints. More precisely, for a pair of distinct points
$\xi,\eta\in\partial T$ and $o$ on the geodesic between $\xi$ and $\eta$, let
$\mu_{\xi,\eta}^{o}$ be the restricted prouniform measure on the set
$\mathcal{J}_{\xi,\eta}^{o}\subset\mathcal{J}$ consisting of all $j$ which
send $0$ to $o$ and the horizontal line in $M$ to the geodesic line
$(\xi,\eta)$.

Note that $\pi_{n}:\mathcal{J}_{\xi,\eta}^{o}\to\partial T$ is not well-
defined anymore. However, we have:

######  Lemma 4.4.

For $n>0$, the measure $(\pi_{n})_{*}\mu_{\xi,\eta}^{o}$ is the normalized
restriction of $\mu_{\partial T}^{o}$ to $E_{n}(\xi)$.

Now, in order to prove Proposition 4.1, we note that if $(\gamma_{n})$ is a
sequence as in Lemma 4.2, then (for $n$ large enough) $\gamma_{n}$ acts as a
translation of length say $N$ on a large segment between $\xi$ and $\eta$. Of
course it is not correct that $\gamma_{n}\pi_{0}(j)=\pi_{n}(j)$ (as we do not
know what happens on the "branches"), but nevertheless it follows that
$\gamma_{n}(\pi_{0})_{*}\mu_{\xi,\eta}^{o}=(\pi_{N})_{*}\mu_{\xi,\eta}^{o}$.

It turns out that this is sufficient to finish the proof of Proposition 4.1.
To do so, we pullback $f$ by $\pi_{0}$ and in fact check weak-* convergence on
$L^{\infty}(\mathcal{J}_{\xi,\eta}^{o})$. Note that having this for one fixed
$o$ is not sufficient, as it only give weak-* convergence on
$L^{\infty}(\Delta,(\pi_{0})_{*}\mu_{\xi,\eta}^{o})$. However since this is
valid for every $o$ we get the convergence with respect to the right measure
class.

By density (Lemma 1.16) it is sufficient to prove the convergence of
$\langle\gamma_{n}f,\phi\rangle$ to $\langle f(\xi),\phi\rangle$ when $\phi$
is the characteristic function of a given embedding of a finite subset of $M$.
But then for $N$ large enough we get that $(\pi_{N})_{*}\phi$ is constant
equal to 1, so that we are precisely reduced to the limit calculated in Lemma
4.3.

###  4.2 Singular flow and contracting sequences

The first step in our proof is to construct sequences of elements of $\Gamma$
with a prescribed dynamic on the boundary. More precisely, our goal is to
prove that there is a sequence of elements contracting almost all of $\Delta$
onto a fixed residue (Proposition 4.5). In fact, it will be later important to
us to also prove that it is possible to do so while also acting as a given
projectivity (Proposition 4.8).

######  Proposition 4.5.

For almost every $(u,v)\in\Delta_{\pm,\mathrm{op}}$, there exists a sequence
$(\gamma_{n})\in\Gamma^{\mathbf{N}}$ such that for every $C\in\Delta$ which is
opposite $u$, the sequence $\gamma_{n}C$ converges to the chamber
$\operatorname{proj}_{v}(\operatorname{proj}_{u}(C))$.

The main tool in the proof of this Proposition is the flow $\mathscr{W}$
constructed in [BCL], whose main properties we recalled in Section 3.2. In
particular, $\mathscr{W}$ is equipped with a measure $\zeta_{\mathscr{W}}$.

######  Lemma 4.6.

For every $(u,v)\in\Delta_{\pm,\mathrm{op}}$, and a finite subset $F\subset
I(u,v)$, the set $E_{F}:=\\{\phi\in\mathscr{W}\mid
F\subset\mathrm{Im}(\phi)\\}$ has positive $\zeta_{\mathscr{W}}$-measure.

###### Proof.

Recall the map $\pi:\mathscr{W}\to\Delta_{\pm,\mathrm{op}}$, associating to an
embedding in $\mathscr{W}$ its two endpoints. By construction of
$\zeta_{\mathscr{W}}$ this map is measure-preserving. Also note that
$F\subset\mathrm{Im}(\phi)$ is equivalent to $F\subset I(\pi(\phi))$, as
$\mathrm{Im}(\phi)$ is in fact equal to $I(\pi(\phi))$ by construction of
$\Upsilon$. Therefore we are led to prove that the set
$E^{\prime}_{F}=\\{(u^{\prime},v^{\prime})\in\Delta_{\pm,\mathrm{op}}\mid
F\subset I(u^{\prime},v^{\prime})\\}$ has a positive
$(\mu_{-}^{o}\times\mu_{+}^{o})$-measure.

Now, using Theorem 3.1, we can define the prouniform measure
$\mu_{\tilde{\mathscr{W}}}$ on the set
$\tilde{\mathscr{W}}=\operatorname{Isom}(\Upsilon,X)$. We can still define a
map $\tilde{\mathscr{W}}\to\Delta_{\pm,\mathrm{op}}$ in the same way, and
still denote it $\pi$. By Corollary 1.11 we get that $\pi$ is measure-class
preserving. On the other hand, by construction, it is clear that the set
$\tilde{E}_{F}=\\{\phi\in\tilde{\mathscr{W}}\mid F\subset\mathrm{Im}(\phi)\\}$
has positive $\mu_{\tilde{\mathscr{W}}}$-measure. Hence $\pi(\tilde{E}_{F})$
has positive $(\mu_{-}^{o}\times\mu_{+}^{o})$-measure. But since
$\pi(\tilde{E}_{F})=E^{\prime}_{F}$ this concludes the proof.

∎

######  Lemma 4.7.

For $m_{\pm}$-almost every $(u,v)\in\Delta_{\pm,\mathrm{op}}$, for every
finite $F\subset I(u,v)$ and $N\in\mathbf{N}$, there exists $n>N$ and
$\gamma\in\Gamma$ such that $\gamma$ acts on $F$ as a translation of length
$n$ in the direction of $u$.

###### Proof.

Let $E_{F}$ be the set of $\phi\in\mathscr{W}$ such that
$F\subset\mathrm{Im}(\phi)$. Let $\phi_{0}\in E_{F}$, and
$B=\phi_{0}^{-1}(F)$. Let $E_{0}=\\{\phi\mid\phi_{|B}=\phi_{0|B}\\}$. Since
there are countably many possible embeddings of $B$ into $X$, up to changing
$\phi_{0}$, we may and shall assume that $\zeta_{\mathscr{W}}(E_{0})>0$. Let
$E$ be the image of $E_{0}$ in $\mathscr{W}/\Gamma$. Note that
$\mathscr{W}/\Gamma$ is equipped with a measure $\zeta_{\mathscr{W}/\Gamma}$
such that the quotient map $\mathscr{W}\to\mathscr{W}/\Gamma$ is measure-class
preserving.

Recall that $\mathscr{W}$ is equipped with an action of a group $S_{\Upsilon}$
translating the $\mathbf{R}$ part of $\Upsilon$, and that this action commutes
with $\Gamma$, hence descends to an action on $\mathscr{W}/\Gamma$. Let $s\in
S_{\Upsilon}$ be the generator, acting by translating the lines in the
direction of the vertex of type $+$. For $N>0$, let
$F_{N}=\\{\varphi\in\mathscr{W}\mid\exists n\geq
N\;\exists\gamma\in\Gamma\;\gamma\circ\varphi\circ s^{n}\in E_{0}\\}$. Note
that $F_{N}$ is $\Gamma$-invariant by construction ; we still denote by
$F_{N}$ its image in $\mathscr{W}/\Gamma$. We claim that $F_{N}$ has full
measure. Indeed, assume by contradiction that its complement $F_{N}^{c}$ has
positive measure. As the action of $S_{\Upsilon}$ on $\mathscr{W}/\Gamma$ is
ergodic [BCL, Theorem 7.3], there exists a $k>N$ such that $s^{k}F_{N}^{c}\cap
E\neq\varnothing$. Therefore there exists $\varphi\in E$ and
$\varphi^{\prime}\in F_{N}^{c}$ with $\varphi^{\prime}s^{k}=\varphi$. In other
words, $\varphi^{\prime}s^{k}\in E_{0}$: this is a contradiction, and
therefore $F_{N}$ has full measure. Therefore the set
$F=\bigcap_{N\in\mathbf{N}}F_{N}$ has full measure in $\mathscr{W}$. The set
of all $(u,v)\in\Delta_{\pm,\mathrm{op}}$ satisfying the condition of the
Lemma is the image of $F$ by the map $\pi$. Since this map is measure-class
preserving, we get the result.

∎

###### Proof of Proposition 4.5.

Fix $(u,v)\in\Delta_{\pm,\mathrm{op}}$ satisfying the condition of Lemma 4.7.
Let $(F_{n})$ be an increasing sequence of finite sets exhausting $I(u,v)$.
Let $\gamma_{n}\in\Gamma$ such that $\gamma_{n}^{-1}$ translates $F_{n}$ on a
length at least $n$ in the direction of $u$.

For $\mu^{o}_{\Delta}$-almost every $C$, the chamber $C$ is opposite $u$. Fix
such a $C$. Let $A_{0}$ be an apartment containing $C$ and $u$, and $A_{1}$ be
an apartment containing $\operatorname{proj}_{u}C$ and $v$. Note that $A_{0}$
and $A_{1}$ both contain a sector pointing towards
$\operatorname{proj}_{u}(C)$. Let $o\in A_{0}\cap A_{1}$ and
$C^{\prime}=\operatorname{proj}_{v}(\operatorname{proj}_{u}C)$. Let $z\in
Q(o,C^{\prime})$, note that $z\in A_{1}$. To prove the desired convergence, it
is enough to prove that for every $n$ large enough we have $z\in
Q(o,\gamma_{n}C)$, that is, $\gamma_{n}^{-1}z\in Q(\gamma_{n}^{-1}o,C)$.

Note that $A_{1}\subset I(u,v)$, so that $o,z\in I(u,v)$. Therefore for $n$
large enough we have $o,z\in F_{n}$, and $\gamma_{n}^{-1}$ will translate
$o,z$ in $A_{1}$ towards $u$. For $n$ large enough it follows that
$\gamma_{n}^{-1}z\in Q(o,\operatorname{proj}_{u}C)$, since
$\operatorname{proj}_{u}C^{\prime}=\operatorname{proj}_{u}C$ by definition.
Let $\rho:A_{0}\to A_{1}$ be the unique isomorphism fixing $A_{0}\cap A_{1}$
pointwise. Since $z\in Q(o,C^{\prime})$, and $\gamma_{n}^{-1}$ translates $o$
and $z$ in $A_{1}$, we have $\gamma_{n}^{-1}z\in
Q(\gamma^{-1}_{n}o,C^{\prime})$. Therefore $\rho(\gamma_{n}^{-1}z)\in
Q(\rho(\gamma_{n}^{-1}o),\rho(C^{\prime}))$. But since
$\gamma_{n}^{-1}o,\gamma_{n}^{-1}z\in Q(o,\operatorname{proj}_{u}C)$ we have
$\rho(\gamma_{n}^{-1}z)=\gamma_{n}^{-1}z$ and
$\rho(\gamma_{n}^{-1}o)=\gamma_{n}^{-1}o$. Also, since $C$ is the chamber of
$A_{0}$ opposite $u$ and at closest distance from $\operatorname{proj}_{u}C$,
and $C^{\prime}$ is the chamber of $A_{1}$ satisfying the same condition, we
get that $\rho(C)=C^{\prime}$. Therefore, we obtain $\gamma_{n}^{-1}z\in
Q(\gamma_{n}^{-1}o,C^{\prime})$, which is the desired result. ∎

In fact, it is possible to obtain a much richer set of sequences, following
any projectivity sequences. Recall that $\operatorname{Aut}(\Upsilon)$ is a
subgroup of the product $\operatorname{Aut}(T)\times\mathbf{Z}$. For every
$(u,v)\in\Delta_{\pm,\mathrm{op}}$, choosing and embedding in $\mathscr{W}$
from $\Upsilon$ to the interval $I(u,v)$ gives us an action of
$\operatorname{Aut}(\Upsilon)$ on $I(u,v)$, and in particular of
$p_{1}^{-1}(\Pi)$ (where $\Pi$ is the projectivity group and $p_{1}$ the first
projection). This action also defines actions of $\Pi$ on
$\operatorname{Res}(u)$ and $\operatorname{Res}(v)$.

######  Proposition 4.8.

For every projectivity $p\in\Pi$, and almost every
$(u,v)\in\Delta_{\pm}^{\mathrm{op}}$, there exists a sequence
$(\gamma_{n})\in\Gamma^{\mathbf{N}}$ such that for every $C\in\Delta$ which is
opposite $u$, the sequence $\gamma_{n}C$ converges to the chamber
$p\circ\operatorname{proj}_{v}(\operatorname{proj}_{u}(C))$.

The main new ingredient in the proof of Proposition 4.8 is the following
lemma, analogous to Lemma 4.7.

######  Lemma 4.9.

Fix a projectivity $p\in\Pi$. For $m_{\pm}$ every
$(u,v)\in\Delta_{\pm,\mathrm{op}}$, for every finite $F\subset I(u,v)$ and
$N\in\mathbf{N}$, there exists $n>N$ and $\gamma\in\Gamma$ such that $\gamma$
acts on $F$ as a a product of the translation of length $n$ in the direction
of $u$, and the projectivity $p$.

###### Proof.

Let $E_{F}$ be the set of $\phi\in\mathscr{F}$ such that
$F\subset\mathrm{Im}(\phi)$. As in Lemma 4.7, let $\phi_{0}\in E_{F}$, and
$B=\phi_{0}^{-1}(F)$. Let $E_{0}=\\{\phi\mid\phi_{|B}=\phi_{0|B}\\}$. Up to
changing $\phi_{0}$ we may and shall assume $\zeta_{\mathscr{W}}(B)>0$.

Since $\tilde{\Pi}$ is closed in $\operatorname{Aut}(T)$, the set
$\tilde{\Pi}_{F}^{p}$ of all $p^{\prime}\in\tilde{\Pi}$ which have the same
action as $p$ on $F$ is open, and therefore its $\tilde{\Pi}$-Haar measure is
positive. Let $E_{p}=\\{\phi\mid\phi_{|B}=\phi_{0|B}\circ p\\}$. By
construction, for almost every $(u,v)\in\Delta_{\pm,\mathrm{op}}$, the
disintegration of $\zeta_{\mathscr{W}}$ along
$\pi:\mathscr{W}\to\Delta_{\pm,\mathrm{op}}$ identifies to the Haar measure on
$\tilde{\Pi}$. For $(u^{\prime},v^{\prime})\in\Delta_{\pm,\mathrm{op}}$,
denote this measure by $\zeta^{u^{\prime},v^{\prime}}_{\mathscr{W}}$.

Now the set of pairs $(u^{\prime},v^{\prime})\in\Delta_{\pm,\mathrm{op}}$ such
that $F\subset I(u^{\prime},v^{\prime})$ has positive $m_{\pm}$-measure. By
the previous discussion, for every such $(u^{\prime},v^{\prime})$, we have
$\zeta^{u^{\prime},v^{\prime}}_{\mathscr{W}}(E_{p})>0$. Integrating this over
all pairs $(u^{\prime},v^{\prime})$ gives that $\zeta_{\mathscr{W}}(E_{p})>0$.

The rest of the argument is exactly similar to Lemma 4.7, replacing $E_{0}$ by
$E_{p}$. ∎

###### Proof of Proposition 4.8.

The argument is almost the same as in the proof of Proposition 4.5. We start
by applying Lemma 4.9 and get a sequence $(\gamma_{n})$ such that
$\gamma_{n}^{-1}$ translates $F_{n}$ in the direction of $u$ by a length $N>n$
and acts as $p^{-1}$ on the transverse tree.

Recall that $\operatorname{Aut}(\Upsilon)$ is a subgroup of the product
$\operatorname{Aut}(T)\times\mathbf{Z}$. We know that $\gamma_{n}^{-1}$ acts
on $F_{n}$ as the pair $(p^{-1},-N)$ for some $n>N$. As in the proof of
Proposition 4.5, let $C$ be a chamber opposite $u$, let $A_{0}$ be an
apartment containing $C$ and $u$. Let
$C^{\prime}=(\operatorname{proj}_{v}(\operatorname{proj}_{u}(C))$ as above,
and $C^{\prime\prime}=pC^{\prime}$ (where $p\in\Pi$ acts on
$\operatorname{Res}(v)$) and let $A_{1}$ be an apartment containing
$\operatorname{proj}_{u}(C)$ and $C^{\prime\prime}$. Note that $A_{1}$ also
contains $C^{\prime}$ since it contains $v$ and $\operatorname{proj}_{u}(C)$.
Let $o\in A_{0}\cap A_{1}$ and let $z\in Q(o,C^{\prime\prime})$.

For $n$ large enough we have $o,z\in F_{n}$ so $\gamma_{n}^{-1}$ will act on
$o,z$ as the pair $(p^{-1},-N)$. By similar arguments as before we obtain that
$\gamma_{n}^{-1}z\in Q(\gamma_{n}^{-1}o,C^{\prime})$, from which it follows
(using again the isomorphism $\rho:A_{0}\to A_{1}$) that $\gamma_{n}^{-1}z\in
Q(\gamma^{-1}_{n}o,C),$ and therefore that $z\in Q(o,\gamma_{n}C)$, which is
the conclusion we wanted.

∎

###  4.3 The detecting flow on the building

Our next goal is to improve Proposition 4.8 into a convergence in weak-*
topology for $L^{\infty}$ functions (Theorem 4.20). It turns out that this is
not a direct argument: in order to do so, we are led to introduce a new flow
(which we call _the detecting flow_), and apply the same kind of recurrence
arguments as in the proof of Proposition 4.8 to this new flow.

We start with a few notations. Start with $\Sigma$ the model apartment of type
$\widetilde{A}_{2}$. In $\Sigma$, we choose a wall $\ell$ that we draw
horizontally. We choose an origin of $\ell$ and index each vertex of $\ell$ by
an integer accordingly (from left to right). For each $n\in\mathbf{Z}$, we
denote by $\ell_{n}$ the line which intersects $\ell$ at the vertex $n$ and
such that the oriented angle $(\ell,\ell_{n})$ is equal to $\pi/3$. For each
$n\in\mathbf{Z}$, we consider a half-plane $\alpha_{n}$ (with an
$\widetilde{A}_{2}$ tessellation) that we glue on $\Sigma$ along $\ell_{n}$.
We let $Y$ be the result of these gluings. Alternatively, we view $Y$ as the
following subset of $\mathbf{R}^{3}$:

$\\{(x,y,z)\mid
z=0\\}\cup\bigcup_{n\in\mathbf{Z}}\\{(x,y,z)\mid\sqrt{3}(x-n)+y=0,~{}z\geq
0\\}$

It is represented on Figure 3 below.

Figure 3: The model space for the detecting flow

######  Lemma 4.10.

The space $Y$ is a convex subset of a wall-tree in an $\widetilde{A}_{2}$
building.

###### Proof.

Fix two opposite vertices $(u^{\prime},v^{\prime})\in\Delta_{\pm,\mathrm{op}}$
and as before let $T_{u^{\prime},v^{\prime}}$ be the tree formed by the set of
geodesics between $u^{\prime}$ and $v^{\prime}$. In
$T_{u^{\prime},v^{\prime}}$ consider a "comb", that is, a graph isomorphic to
the space $M$ as in Figure 2 (or in other word, a geodesic line $g$ with
attached geodesic rays starting from each vertex of $g$). The union of all
geodesics of $X$ corresponding to points of $M$ is then isomorphic to $Y$. ∎

In particular, $Y$ is ind-$P_{\Upsilon}$, where $P_{\Upsilon}$ is the property
of being isometric to a convex subset of a model wall tree $\Upsilon$, as
defined in the beginning of §3. Furthermore, the space $X$ is
$P_{\Upsilon}$-symmetric, by Theorem 3.1. Hence we can define the Radon
prouniform measure on $\operatorname{Isom}(Y,X)$.

######  Definition 4.11.

The _detecting flow_ is the space $\mathcal{I}=\operatorname{Isom}(Y,X)$,
endowed with the prouniform measure, denoted $\mu_{\mathcal{I}}$.

The space $\mathcal{I}$ is endowed with an action of $\operatorname{Aut}(X)$
(by post-composition), and a ’shift’ $S_{Y}:\mathcal{I}\to\mathcal{I}$ which
acts on $Y$ by translating $\Sigma$ and sending $\alpha_{n}$ on
$\alpha_{n+1}$.

Furthermore, let $C_{+}$ be the Weyl chamber of $\Sigma$, bounded by $\ell$
and one of the lines $\ell_{i}$, and $C_{-}$ be its opposite in $\Sigma$.
Similarly let $C_{n}$ be the Weyl chamber of the half-plane $\alpha_{n}$ which
is not bounded by $\ell_{n}$. We define $\pi_{+}:\mathcal{I}\to\Delta$ (resp.
$\pi_{-}$, $\pi_{n}$) by $\pi_{+}(f)=f(C_{+})$ (resp. $\pi_{-}(f)=f(C_{-})$,
$\pi_{n}(f)=f(C_{n})$).

Let $Y_{k}$ be the ball of radius $k$ in $Y$, and
$\mathcal{I}_{k}=\operatorname{Isom}(Y_{k},X)$. Since $Y_{k}$ is finite, the
space $\mathcal{I}_{k}$ is equipped with the uniform measure $\mu_{Y_{k}}$.
Let $r_{k}:\mathcal{I}\to\mathcal{I}_{k}$ be the restriction. We view
$L^{1}(\mathcal{I}_{k},\mu_{Y_{k}})$ as a subset of
$L^{1}(\mathcal{I},\mu_{Y})$ by identifying it with its pullback by
$(r_{k})^{*}$.

For $g\in\mathcal{I}$, the line $g(\ell)\subset g(\Sigma)$ joins two points
$(u,v)\in\Delta_{{\pm,\mathrm{op}}}$. This defines a map
$p:\mathcal{I}\to\Delta_{{\pm,\mathrm{op}}}$. The pushforward of
$\mu_{\mathcal{I}}^{o}$ by this map is a prouniform measure that we denote
$\mu_{{\pm,\mathrm{op}}}^{o}$.

Using Lemma 1.10, we get that for every $(u,v)\in\Delta_{\pm,\mathrm{op}}$,
and $o\in I(u,v)$, we have a probability measure $\mu_{u,v}^{o}$ on
$p^{-1}(u,v)$ which is the restricted prouniform measure on the set of
embeddings with $\ell$ sent to the line passing through $o$ and joining $u$ to
$v$, and these measures satisfy the disintegration formula

$\mu_{\mathcal{I}}^{o}=\int_{\Delta_{\pm,\mathrm{op}}}\mu_{u,v}^{o}d\mu_{\pm,\mathrm{op}}^{o}(u,v)$

In the sequel, we will denote $p^{-1}(u,v)$ by $\mathcal{I}^{u,v}$.

Recall the definition of $E_{n}(v)$ from §3.2.2.

######  Proposition 4.12.

Let $(u,v)\in\Delta_{\pm}^{\mathrm{op}}$, and $o\in I(u,v)$. The measure
$(\pi_{0})_{*}\mu_{u,v}^{o}$ is absolutely continuous with respect to
$\mu_{\Delta}^{o}$. Furthermore for any $n>0$ the measure
$(\pi_{n})_{*}\mu_{u,v}^{o}$ is equal to the normalization of the restriction
of $\mu_{\Delta}^{o}$ to $E_{n}(v)$.

Write $o,x_{1},x_{2},\dots$ (resp. $o,y_{1},y_{2},\dots$) for the consecutive
vertices of the geodesic ray $[ou)$ (resp. $[ov)$).

######  Definition 4.13.

A Weyl chamber $C$ is said to be _$(u,v)$ -generic_ if it is opposed to both
$u$ and $v$.

######  Lemma 4.14.

Let $Q=Q(o,C)$ be a sector of $X$ with basepoint $o$, and assume that the
alcove of $Q$ containing $o$ is opposite (in $\operatorname{lk}(o)$) to both
$x_{1}$ and $y_{1}$. Then $C\in\pi_{0}(\mathcal{I}_{u,v}^{o})$.

A Weyl chamber $C$ such that $Q(o,C)$ satisfies the condition of Lemma 4.14
will be said to be _$(u,v)$ -generic at $o$_.

######  Remark 4.15.

Since $\mathcal{I}_{u,v}^{o}$ is the set of embeddings containing the geodesic
line from $u$ to $v$ passing through $o$, this set does not change when
choosing a different origin. In particular for $n>0$ we have
$C\in\pi_{0}(\mathcal{I}_{u,v}^{x_{n}})$ if and only if $C$ is $(u,v)$-generic
at $x_{n}$. Applying the shift $S_{\Upsilon}$, this implies that
$C\in\pi_{n}(\mathcal{I}_{u,v}^{o})$ if and only if $C$ is $(u,v)$-generic at
$x_{n}$.

###### Proof.

From Corollary 2.4 we know that if $\alpha$ is a half-apartment, and $c$ is an
alcove intersecting $\partial\alpha$, then $c\cup\alpha$ is contained in an
apartment. Using this fact, one can prove by induction that there exists an
apartment $A_{n}$ containing $Q\cup[o,x_{n}]$. Therefore, by Corollary 2.3
there exists an apartment $A$ containing $Q\cup[o,u)$. Let $\alpha$ be the
half-apartment of $A$ containing $Q$, bounded by a wall passing through $o$
but not adjacent to $Q$. Let $\alpha^{\prime}$ be its complement in $A$, so
that $\alpha^{\prime}$ contains $[ou)$.

Using again Corollary 2.4, we see by induction that there exists an apartment
$A^{\prime}_{n}$ containing $\alpha\cup[o,y_{n}]$, and therefore there exists
an apartment $A^{\prime}$ containing $\alpha\cup[o,v)$. Let $\beta$ be the
complement of $\alpha$ in $\alpha^{\prime}$, so that $[o,v)\subset\beta$. Note
that $\beta$ does not intersect $\alpha^{\prime}$, since $x_{1}$ and $y_{1}$
are opposite in $\operatorname{lk}(o)$. Since $\beta$ is a half-apartment
bounded by $\partial\alpha^{\prime}=\partial\alpha$, with empty intersection
with $\alpha^{\prime}$, it follows that $\beta\cup\alpha^{\prime}$ is also an
apartment.

Therefore $\alpha\cup\alpha^{\prime}\cup\beta$ is isometric to a convex
subspace $Y^{\prime}$ of $Y$. Fixing such an isometry
$f^{\prime}:Y^{\prime}\to\alpha\cup\alpha^{\prime}\cup\beta$, one can extend
$f^{\prime}$ to an isometric embedding $f^{\prime}:Y\to X$, which satisfies
$Q=\pi_{0}(f)$.

∎

###### Proof of Proposition 4.12.

Let $E$ be the clopen subset of $\Delta$ defined by the set of classes of
sector satisfying the conditions of Lemma 4.14. We first prove that $\mu^{o}$
and $(\pi_{0})_{*}\mu_{u,v}^{o}$ are equivalent in restriction to $E$. Since
$(\pi_{0})_{*}\mu_{u,v}^{o}$ is zero outside of $E$, this will prove the first
part of the Proposition.

Recall that $\Lambda\subset\Sigma$ is a model sector (based at $0$), so that
$\mu^{o}_{\Delta}$ is the prouniform measure on the set of embeddings
$\operatorname{Isom}(\Lambda,X)$ which send $0$ to $o$. Let $c$ be the alcove
of $\Lambda$ containgin $0$. Let $\Lambda^{\prime}\subset\Lambda$ be a finite
convex subcomplex containing $c$, and let
$\alpha\in\operatorname{Isom}(\Lambda^{\prime},X)$ sending $0$ to $o$. By
definition, $\mu_{\Delta}^{o}(\operatorname{Isom}(\Lambda,X)^{\alpha})$ is
independent of $\alpha$, equal to a constant say $K$. Now, view $\Lambda$ as
the sector of $Y$ based at $0$ and defining $\pi_{0}$. Say that $\alpha$ is
_admissible_ if $\alpha(c)$ is opposite to both $x_{1}$ and $y_{1}$. By Lemma
4.14, we have
$\pi_{0}(\operatorname{Isom}(Y,X)^{\alpha}\cap\mathcal{I}_{u,v}^{o})$ which is
either empty or equal to $\pi_{0}(\operatorname{Isom}(Y,X)^{\alpha})$,
depending on whether $\alpha(c)$ is admissible or not. It follows from
Corollary 1.11 that
$(\pi_{0})_{*}\mu_{u,v}^{o}(\operatorname{Isom}(Y,X)^{\alpha})$ is either
equal to $0$ (if $\alpha$ is not admissible) or to some constant independent
of $\alpha$ (if it is). Thus, we have
$(\pi_{0})_{*}\mu_{u,v}^{o}(\operatorname{Isom}(Y,X)^{\alpha})=K^{\prime}$
where $K^{\prime}$ is the proportion of admissible $\alpha$, amongst all the
possible ones which send $0$ to $o$.

Since $K$ does not depend on $\Lambda^{\prime}$, we deduce that
$\frac{d(\pi_{0})_{*}(\mu_{u,v}^{o})}{d\mu^{o}_{\Delta}}(C)=0$ if $C$ is not
$(u,v)$-generic at $o$, and $\frac{K^{\prime}}{K}$ if it is. This proves that
on the clopen set of chambers which are $(u,v)$-generic at $o$,
$(\pi_{0})_{*}\mu_{u,v}^{o}$ and $\mu^{o}_{\Delta}$ are equivalent.

Now let us prove the second statement. We claim first that for $n>0$,
$E_{n}(v)$ is exactly the set of chambers which are $(u,v)$-generic at
$x_{n}$. Indeed, if $C\in E_{n}(v)$, then $Q(o,C)$ contains the geodesic ray
$[ox_{n}]$, and it follows that $C$ is opposite $v$ at $x_{n}$. The ray
$[x_{n}\operatorname{pr}_{+}(C)]$ starts with a segment of the same type as
$[x_{n}x_{n+1}]$, but which is not equal to this segment. Hence these two
segments are opposite in $\operatorname{lk}(x_{n})$, which proves that $C$ is
$(u,v)$-generic at $x_{n}$. Conversely, if $C$ is $(u,v)$-generic at $x_{n}$,
then the union of the rays $[ox_{n}]\cup[x_{n}\operatorname{pr}_{+}(C))$ has
an angle $\pi$ at $x_{n}$, so is still a geodesic ray. Hence
$C\in\Omega_{o}(\lambda_{n}(v))$. Furthermore it is clear that $x_{n+1}\not\in
Q(o,C)$, hence we do have $C\in E_{n}(v)$.

Therefore $(\pi_{n})_{*}\mu$ is supported on $E_{n}(v)$. Using the same
argument as for $\pi_{0}$, we see that
$\frac{d(\pi_{n})_{*}\mu_{u,v}^{o}}{d\mu_{\Delta}^{o}}$ is in fact constant
(and $>0$) on $E_{n}(v)$. It follow that $(\pi_{n})_{*}\mu_{u,v}^{o}$ is equal
to a constant times the restriction of $\mu_{u,v}^{o}$ on $E_{n}(v)$ ; since
it is a probability measure we get the result.

∎

Finally, we are interested in what happens when we change the basepoint $o$.

######  Lemma 4.16.

For any $(u,v)$-generic Weyl chamber $C$, there exists $o$ between $u$ and $v$
such that $C$ is $(u,v)$-generic at $o$.

###### Proof.

Let $\ell$ be a geodesic line between $u$ and $v$. Let us number the vertices
of $\ell$ by $(\ell_{i})_{i\in\mathbf{Z}}$ in such a way that
$\lim_{i\to+\infty}\ell_{i}=v$. There exists an apartment containing $u$ and
$C$ and every $\ell_{i}$, for $i$ small enough. However, if for every
$i\in\mathbf{Z}$ there existed an apartment containing $(u\ell_{i}]$ and $C$,
then there would be an apartment containing $\ell$ and $C$. This is not
possible as $C$ is opposite both $u$ and $v$.

It follows that there exists a maximal $i_{0}$ such that $u,\ell_{i_{0}}$ and
$C$ are in a common apartment. We define $o=\ell_{i_{0}}$ and prove that $C$
is $(u,v)$-generic at $o$. By construction $o$ is in some apartment containing
$u$ and $C$, and $u$ is opposite $C$, so the projection of $C$ and $u$ on
$\operatorname{lk}(o)$ are opposite.

On the other hand, the geodesic ray $[o,v)$ is the line
$\ell_{i_{0}},\ell_{i_{0}+1},\dots$. Let $y=\ell_{i_{0}+1}$ Let $c$ be the
alcove which is the projection of $C$ on $\operatorname{lk}(o)$ and $x$ be the
vertex of $c$ of the same type as $\ell_{i_{0}-1}$. We want to prove that $x$
is opposite $y$ in $\operatorname{lk}(o)$.

Assume not. Since $\operatorname{lk}(o)$ is of type $A_{2}$, and $x$ and $y$
have distinct type and are not opposite, they are adjacent. Start with an
apartment $A_{0}$ containing $u$, $o$ and $C$ (which exists by assumption). By
construction, $x$ is also a vertex in $A_{0}$. Let $\hat{h}$ be the wall of
$A_{0}$ containing $[ux)$, and $h$ be the half-space of $A_{0}$ containing
$C$. Since $y$ is adjacent to $y$, we see that $y$ is a vertex which is
contained in an alcove which has an edge in $\hat{h}$. Therefore by Corollary
2.4 there exists an apartment $A$ containing $h$ and $y$. In particular, $A$
contains $u,y$ and $C$. This contradicts the maximality of $i_{0}$.

∎

######  Proposition 4.17.

The sum over all $o\in I(u,v)$ of the measure classes of
$(\pi_{0})_{*}\mu_{u,v}^{o}$ is equivalent to $\mu_{\Delta}^{o}$.

######  Remark 4.18.

The measure $\sum_{o\in I(u,v)}(\pi_{0})_{*}\mu_{u,v}^{o}$ is locally
infinite. However, the measure class is well-defined.

###### Proof.

By the previous proposition this sum is absolutely continuous with respect to
$\mu^{o}_{\Delta}$. In fact it also shows that it is equivalent to
$\mu^{o}_{\Delta}$ in restriction to the union of sets of chambers which are
$(u,v)$ generic at $o$ for some $o$. By Lemma 4.16, this set is in fact the
set of all $(u,v)$-generic Weyl chambers, which is of full measure. Hence the
two measure classes are equal. ∎

###  4.4 Contracting properties

As explained above, the goal of the detecting flow is to improve on
Proposition 4.8 to a convergence for bounded measurable functions. This is
what we finally do in this section: our goal is to prove Theorem 4.20 below.

######  Lemma 4.19.

For every $f\in L^{\infty}(\Delta)$, for $\mu_{\pm,\mathrm{op}}$-almost every
$(u,v)\in\Delta_{\pm,\mathrm{op}}$, for every $o$ between $u$ and $v$ and for
every $\varphi\in L^{1}(\mathcal{I}_{u,v}^{o})$ we have

$\int\varphi(y)f(\pi_{n}y)d\mu_{u,v}^{o}(y)\longrightarrow\int\varphi(y)\pi_{\infty}^{*}(f)d\mu_{u,v}^{o}(y)$

###### Proof.

Since the norm of the function $y\mapsto f(\pi_{n}y)-\pi_{\infty}^{*}(f)$ is
bounded by $2\|f\|$, it is enough to prove the convergence with respect to a
total subset of functions $\varphi$. Hence by Lemma 1.16, it is sufficient to
prove the Lemma for $\varphi$ in the pullback of $L^{1}(\mathcal{I}_{k})$, for
every $k>0$. Fix such a $k$. By linearity it is also sufficient to prove the
Lemma when $\varphi$ is the characteristic function of a fixed embedding
$\iota_{0}\in\mathcal{I}_{k}\cap\mathcal{I}_{u,v}^{o}$. Fix
$\iota\in\mathcal{I}_{u,v}^{o}$ such that $\iota_{|\Upsilon_{k}}=\iota_{0}$
and let $\xi=\pi_{\infty}(\iota)$.

Fix $n>k$. Note that there exists an automorphism $\tau$ of order 2 of $Y$
which exchanges the two half-apartments bounded by $\ell_{n}$ not containing
$0$ ; in particular $\pi_{n}\circ\tau=\pi_{\infty}$. Furthermore $\tau$ fixes
pointwise the ball of radius $k$ in $Y$.

Let $\eta\in E_{n}(v)$. Then $(\pi_{n})_{*}\varphi(\eta)=1$ if and only if
there exists an extension $\iota^{\prime}\in\mathcal{I}_{u,v}^{o}$ of
$\iota_{0}$ such that $\pi_{n}(\iota^{\prime})=\eta$. We claim that
$(\pi_{n})_{*}\varphi(\eta)=1$ if and only if $\lambda_{k,n}(\xi)\in
Q(o,\eta)$. Indeed, if there is an extension $\iota^{\prime}$ as above, then
let $\xi^{\prime}=\pi_{\infty}(\iota^{\prime})$ (and note that
$\operatorname{pr}_{+}(\xi^{\prime})=v$). Since $\iota^{\prime}$ is equal to
$\iota$ in restriction to $Y_{k}$ (which contains $\lambda_{k,0}$) we have
$\lambda_{k,0}(\xi^{\prime})=\lambda_{k,0}(\xi)$, and the geodesic ray from
$\lambda_{k,0}(\xi)$ to $u$ is contained in $Q(o,\xi^{\prime})$. Applying
$\tau$ we see that the initial segment of length $n$ of this ray is contained
in $Q(o,\pi_{n}(\iota))$, so that indeed $\lambda_{k,n}(\xi)\in Q(o,\eta)$.
Conversely, assume that $\lambda_{k,n}(\xi)\in Q(o,\eta)$. Then we have
$Q(\lambda_{k,n}(\xi),\eta)\subset Q(o,\eta)$, and this sector is also
contained in a half-apartment $\alpha$ which is bounded by the wall
$\iota(\ell_{n})$. Replacing the half-apartment of $\iota$ bounded by
$\ell_{n}$ by $\alpha$, we still get an isometric embedding of $Y$ which is
equal to $\alpha$ on $Y_{k}$.

Using the claim, together with Proposition 4.12 and Lemma 3.26 we get

$\int\varphi(y)f(\pi_{n}y)d\mu_{u,v}^{o}(y)=\frac{1}{\mu(E_{n}(v))}\int_{E_{k,n}(\xi)}f(\eta)d\mu_{\Delta}^{o}(\eta)$

Note also that for $k>0$ we have
$\mu_{\Delta}^{o}(E_{k,n}(\xi))=\frac{1}{(q+1)q^{k-1}}\mu_{\Delta}^{o}(E_{n}(v))$.
Hence by Lemma 3.25 we get that

$\lim_{n}\int\varphi(y)f(\pi_{n}y)d\mu_{u,v}^{o}(y)=\frac{1}{(q+1)q^{k-1}}\frac{1}{\mu_{u}^{o}(\Omega_{k,\infty}(\xi))}\int_{\Omega_{k,\infty}(\xi)}f(\eta)d\mu_{u}^{o}(\eta)$

On the other hand, we have
$\mu_{u}^{o}(\Omega_{k},\infty)(\xi))=\frac{1}{(q+1)q^{k-1}}$. Also, note that
$(\pi_{n})_{*}\varphi(\eta)=1$ if and only if $\eta\in\Omega_{k,\infty}(\xi)$.
This implies that

$\int\varphi(y)\pi_{\infty}^{*}(f)d\mu_{u,v}^{o}(y)=\int_{\Omega_{k,\infty}(\xi)}f(\eta)d\mu_{u}^{o}(\eta).$

The result follows. ∎

(Note that the "almost sure" set of pairs $(u,v)$ for which Lemma 4.19 hold
might depend on $f$.)

This finally allows us to prove the main theorem of this section.

######  Theorem 4.20.

For every $f\in L^{\infty}(\Delta)$, for almost every
$(u,v)\in\Delta_{\pm}^{\mathrm{op}}$, for every projectivity $p$ (acting on
$\operatorname{Res}(v)$), there is a sequence $(\gamma_{n})_{n\in\mathbf{N}}$
of elements of $\Gamma$ such that the sequence
$(\gamma_{n}.f)_{n\in\mathbf{N}}$ converges (in the weak-* topology on
$L^{\infty}(\Delta)$) to the map $x\mapsto
f(p\circ\operatorname{proj}_{v}(\operatorname{proj}_{u}(x)))$.

###### Proof.

Let us fix $f\in L^{\infty}(\Delta)$. Fix $(u,v)$ in the intersection of the
full measure sets provided by Corollary 4.19 and Lemma 4.9. Let $(\gamma_{n})$
be the sequence provided by Lemma 4.9.

Note that if $g\in\mathcal{I}_{u,v}$ then
$\pi_{\infty}(g)=\operatorname{proj}_{v}\circ\operatorname{proj}_{u}\circ\pi_{0}(g)$:
this follows from the construction of the detecting flow. In particular by
Proposition 4.8 we have $(\gamma_{n}\pi_{0}(g))_{n\in\mathbf{N}}$ which
converges to $p\circ\pi_{\infty}(g)$.

We claim that the sequence $(\gamma_{n}^{*}\pi_{0}^{*}f)$ converges to
$(p\circ\pi_{\infty})^{*}f$ in the weak-* topology of
$L^{\infty}(\mathcal{I}_{u,v}^{o})$, for every $o$ in $I(u,v)$. Granting the
claim, this proves that $\gamma_{n}f$ weak-* converges to $x\mapsto
f(p\circ\operatorname{proj}_{v}(\operatorname{proj}_{u}(x)))$ in the weak-*
topology of $L^{\infty}(\Delta,(\pi_{0})_{*}\mu_{u,v}^{o})$. Since this is
valid for every $o$, using Proposition 4.17 the same convergence holds in
$L^{\infty}(\Delta,\mu_{\Delta})$, which is the conclusion of the Theorem.

So it remains to prove the claim. Fix $o\in I(u,v)$. By compactness of balls
in $L^{\infty}(\mathcal{I}_{u,v}^{o})$ (for the weak-* topology), it is enough
to prove that every convergent subsequence of $(\gamma_{n}^{*}\pi_{0}^{*}f)$
converges to $(p\circ\pi_{\infty})^{*}f$. Therefore, we may and shall assume
that the sequence $(\gamma_{n}^{*}\pi_{0}^{*}f)$ converges. In order to
identify the limit, it is sufficient to identify the limit against a dense set
of functions in $L^{1}(\mathcal{I}_{u,v}^{o})$. This dense set will be the
union of $L^{1}(\mathcal{I}_{u,v}^{o,k})$ for $k\geq 0$ (see Lemma 1.16). In
fact, by linearity, it is enough to identify the limit for the characteristic
function $\varphi$ of a fixed embedding $\iota\in\mathcal{I}_{u,v}^{o,k}$.

So fix such a $k$, and $g\in\mathcal{I}_{u,v}^{o,k}$. For $n$ large enough, we
know that $\gamma_{n}$ acts as the composition of a translation of the ball of
radius $k$ in $I(u,v)$ by some length $N$ with $p$, where $N=N(n)$ tends to
$+\infty$ as $n$ increases. Recall that we define the shift
$S_{Y}:\mathcal{I}\to\mathcal{I}$, acting on $Y$ by precomposition (sending
the line $\ell_{n}$ to $\ell_{n+1}$).

Note that $\gamma_{n}S_{Y}^{-N}$ stabilizes $\mathcal{I}^{o,k}_{u,v}$. Since
this set is finite, we may and shall extract a subsequence again such that for
every $n$ we have $\gamma_{n}S_{Y}^{-N}\iota$ which is constant, equal to say
$\iota^{\prime}$. Let $\varphi^{\prime}\in L^{1}(\mathcal{I}_{u,v}^{o,k})$ be
the characteristic function of $\iota^{\prime}$. Note that
$\iota^{\prime}=p\circ\iota$ in restriction to $I(u,v)$, by construction of
the sequence $(\gamma_{n})$.

We claim that
$\langle(p\circ\pi_{\infty})^{*}f,\varphi\rangle=\langle\pi_{\infty}^{*}f,\varphi^{\prime}\rangle$
(where $\langle\cdot,\cdot\rangle$ means integration relative to
$\mu_{u,v}^{o}$). Indeed, note that since $p\circ\iota=\iota^{\prime}$ on
$I(u,v)$ we have
$(p\circ\pi_{\infty})_{*}\varphi=(\pi_{\infty})_{*}\varphi^{\prime}$.

Now we get that

$\displaystyle\langle\gamma_{n}^{*}\pi_{0}^{*}f,\varphi\rangle$
$\displaystyle=\langle f,(\gamma_{n}\pi_{0})_{*}\varphi\rangle$
$\displaystyle=\langle f,(\pi_{0}\gamma_{n})_{*}\varphi\rangle$
$\displaystyle=\langle f,\pi_{0}S^{N}\varphi^{\prime}\rangle$
$\displaystyle=\langle f,\pi_{N}\varphi^{\prime}\rangle$
$\displaystyle=\langle\pi_{N}^{*}f,\varphi^{\prime}\rangle$

which converges to $\langle\pi_{\infty}^{*}f,\varphi^{\prime}\rangle$ by
Corollary 4.19. Using the claim above, this limit is also equal to $\langle
p\circ\pi_{\infty}^{*}f,\varphi\rangle$. This concludes the proof.

∎

Let us emphasize the particular case when $p$ is the identity:

######  Corollary 4.21.

For every $f\in L^{\infty}(\Delta)$, for almost every
$(u,v)\in\Delta_{\pm}^{\mathrm{op}}$, there is a sequence
$(\gamma_{n})_{n\in\mathbf{N}}$ of elements of $\Gamma$ such that the sequence
$(\gamma_{n}.f)_{n\in\mathbf{N}}$ converges (in the weak-* topology on
$L^{\infty}(\Delta)$) to the map $x\mapsto
f(\operatorname{proj}_{v}(\operatorname{proj}_{u}(x)))$.

###  4.5 Dynamics on the boundary

Recall that an action of a group $G$ on a topological space $Z$ is
_topologically transitive_ if for every non-empty open sets $U,V\subset Z$,
there exists $g\in G$ such that $gU\cap V\neq\varnothing$. If $Z$ is Polish,
it is equivalent to the existence of a dense orbit in $Z$. The action is
_minimal_ if every orbit is dense.

We say that the action is _topologically 2-minimal_ if the orbit of every pair
of distinct points is dense in $Z\times Z$.

The main interest of topological 2-minimality is the following.

######  Lemma 4.22.