# Strong transitivity, Moufang’s condition
and the Howe–Moore property

Corina Ciobotaru<EMAIL_ADDRESS>
(October 8, 2021)

###### Abstract

Firstly, we prove that every closed subgroup $H$ of type-preserving
automorphisms of a locally finite thick affine building $\Delta$ of dimension
$\geq 2$ that acts strongly transitively on $\Delta$ is Moufang. If moreover
$\Delta$ is irreducible and $H$ is topologically simple, we show that $H$ is
the subgroup $\operatorname{\mathbb{G}}(k)^{+}$ of the $k$-rational points
$\operatorname{\mathbb{G}}(k)$ of the isotropic simple algebraic group
$\operatorname{\mathbb{G}}$ over a non-Archimedean local field $k$ associated
with $\Delta$. Secondly, we generalise the proof given in [BM00b] for the case
of bi-regular trees to any locally finite thick affine building $\Delta$, and
obtain that any topologically simple, closed, strongly transitive and type-
preserving subgroup of $\mathrm{Aut}(\Delta)$ has the Howe–Moore property.
This proof is different than the strategy used so far in the literature and
does not relay on the polar decomposition $KA^{+}K$, where $K$ is a maximal
compact subgroup, and the important fact that $A^{+}$ is an abelian maximal
sub-semi-group.

##  1 Introduction

In this article we will work in the setting where $\Delta$ is a locally finite
thick affine building of dimension $\geq 2$ that will always be considered
with its complete system of apartments. With respect to that, the ideal
boundary (also called the visual boundary) of $\Delta$ is denoted by
$\partial_{\infty}\Delta$ and this is endowed with a structure of a spherical
building. By [AB, Theorem 11.79] the apartments of $\partial_{\infty}\Delta$
are in one to one correspondence with those of $\Delta$. We denote by
$\mathrm{Aut}(\Delta)$ the full group of automorphisms of $\Delta$ and by
$\mathrm{Aut}_{0}(\Delta)\leq\mathrm{Aut}(\Delta)$ the subgroup of all type-
preserving automorphisms, meaning those automorphisms that preserve a chosen
colouring of the vertices of the model chamber and thus of the vertices of the
chambers of $\Delta$.

We say $G\leq\mathrm{Aut}_{0}(\Delta)$ acts strongly transitively on $\Delta$
if for any two pairs $(\mathcal{A}_{1},c_{1})$ and $(\mathcal{A}_{2},c_{2})$,
consisting each of an apartment $\mathcal{A}_{i}$ of $\Delta$ and a chamber
$c_{i}\in\mathrm{Ch}(\mathcal{A}_{i})$, there exists $g\in G$ such that
$g(\mathcal{A}_{1})=\mathcal{A}_{2}$ and $g(c_{1})=c_{2}$.

######  Definition 1.1.

Let $X$ be an affine or spherical building. For a simplex $\sigma\subset X$
let $\operatorname{St}_{X}(\sigma)$ be the set of all simplices in $X$ that
contain $\sigma$ as a sub-simplex, $\operatorname{St}_{X}(\sigma)$ is called
the star of $\sigma$ in $X$. A root $\alpha$ of $X$ is a half-apartment of $X$
and its boundary wall (not the ideal boundary) is denoted by $\partial\alpha$.
Let $\mathcal{A}_{X}(\alpha)$ be the set of all apartments of $X$ that contain
$\alpha$ as a half-apartment.

######  Definition 1.2.

Let $H$ be a closed subgroup of $\mathrm{Aut}_{0}(\Delta)$ and $\alpha$ any
root of $\partial_{\infty}\Delta$. The root group $U_{\alpha}(H)$ associated
with $\alpha$ and $H$ is defined to be the set of all elements $g$ of $H$
which fix $\operatorname{St}_{\partial_{\infty}\Delta}(P)$ pointwise for every
panel $P$ in $\alpha-\partial\alpha$. We take

$H^{+}:=\langle U_{\alpha}(H)\;|\;\alpha\text{ root of
}\partial_{\infty}\Delta\rangle.$

######  Remark 1.3.

Let $H$ be a closed subgroup of $\mathrm{Aut}_{0}(\Delta)$ and $\alpha$ a root
of $\partial_{\infty}\Delta$. Then, as the action of $H$ on
$\partial_{\infty}\Delta$ is continuous, we have $U_{\alpha}(H)$ is a closed
subgroup of $H$.

######  Definition 1.4.

Let $H$ be a closed subgroup of $\mathrm{Aut}_{0}(\Delta)$. We say $H$ is
Moufang if for every root $\alpha$ of $\partial_{\infty}\Delta$ the associated
root group $U_{\alpha}(H)$ acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$.

Recall the following well known results from the literature. For
$\operatorname{\mathbb{G}}$ a semisimple algebraic group over a non-
Archimedean local ﬁeld $k$, of $k$-rank $\geq 2$ (i.e. isotropic over $k$ (see
[Bo, Definition 20.1])), let $G:=\operatorname{\mathbb{G}}(k)$ be the group of
all $k$–rational points of $\operatorname{\mathbb{G}}$. By [BT, BTII], with
the group $G$ one associates a locally ﬁnite thick affine building $\Delta$
where $G$ acts by type-preserving automorphisms and strongly transitively.
Moreover, by [BoT65], the group $G$ always possesses a canonical BN-pair, and
if moreover the spherical building $\partial_{\infty}\Delta$ at infinity of
$\Delta$ is irreducible and of dimension $\geq 2$, then
$\partial_{\infty}\Delta$ is Moufang (in the sense of [AB, Definition 7.27],
see [AB, Theorem 7.59]). Also, by [BoT, Prop.6.14], or [Ma, Theorem. 2.3.1],
$G^{+}=\langle U_{\alpha}(G)\;|\;\alpha\text{ is a root of
}\partial_{\infty}\Delta\rangle$ is a closed normal subgroup of $G$ and the
factor group $G/G^{+}$ is compact.

If in addition we suppose $\operatorname{\mathbb{G}}$ is also (absolutely)
almost simple (for the definition see [Ma, page 21], thus $\Delta$ is
irreducible by [Bo, Proposition 14.10(3)]), we will call
$\operatorname{\mathbb{G}}$ an isotropic simple algebraic group over the non-
Archimedean local ﬁeld $k$. By [BoT, Remark 6.15], if moreover
$\operatorname{\mathbb{G}}$ is simply connected (see [Ma, Definition 1.4.9]),
then $G=G^{+}$.

Let $\operatorname{\mathbb{G}}$ be a semisimple, (absolutely) almost simple
algebraic group over a local field $k$. Then by [Tits74, Section 5.8] the
spherical building $\partial_{\infty}\Delta$ associated with
$G=\operatorname{\mathbb{G}}(k)$ (we keep the same notation as above)
determines $k$ up to isomorphism, $G^{+}$ up to isomorphism and
$\operatorname{\mathbb{G}}$ up to special isogeny. If moreover
$\operatorname{\mathbb{G}}$ is $k$-split then those groups are classified up
to isogeny by the type of the root system, and each isogeny class contains a
unique (up to isomorphism) simply connected group and a unique adjoint group
(see [Tits79, Section 4], [RR, Section 2.2]).

The first result that we prove in this article is the following:

######  Theorem 1.5.

Let $\Delta$ be a locally finite thick affine building of dimension $\geq 2$.
Let $H$ be a closed subgroup of $\mathrm{Aut}_{0}(\Delta)$ that acts strongly
transitively on $\Delta$. Then $H$ is Moufang.

To prove Theorem 1.5 we follow the same steps and techniques as in [CM1,
Section 7.G, Theorem D] where the case $H=\mathrm{Aut}_{0}(\Delta)$ was
treated.

By [CM, Theorem 1.1 and Theorem 1.2 ] one can notice that a closed and
strongly transitive subgroup of $\mathrm{Aut}_{0}(\Delta)$ is necessarily non-
compact.

Following Weiss [Wi, Chapter 28] and the remark from [CM1, Section 3.B] we
recall the following non-trivial result:

######  Theorem 1.6.

Let $\Delta$ be an irreducible locally finite thick affine building of
dimension $\geq 2$ whose building at infinity $\partial_{\infty}\Delta$ is
Moufang (in the sense of [AB, Definition 7.27]). Then $\Delta$ is precisely a
Bruhat–Tits building associated with the group $\operatorname{\mathbb{G}}(k)$
of an isotropic simple algebraic group $\operatorname{\mathbb{G}}$ over a non-
Archimedean local field $k$, of $k$-rank $\geq 2$.

The second result of the article, that uses Theorem 1.5 above, is the
uniqueness of topologically simple, closed and strongly transitive subgroups
$H\leq\mathrm{Aut}_{0}(\Delta)$, where $\Delta$ is an irreducible locally
finite thick affine buildings of dimension $\geq 2$. Those subgroups are in
fact the groups $G^{+}=\operatorname{\mathbb{G}}(k)^{+}$ associated with the
isotropic simple algebraic groups $\operatorname{\mathbb{G}}$ over non-
Archimedean local fields $k$, and so of algebraic origin.

######  Theorem 1.7.

Let $\Delta$ be an irreducible locally finite thick affine building of
dimension $\geq 2$ such that $\mathrm{Aut}_{0}(\Delta)$ acts strongly
transitively on $\Delta$. Then the only topologically simple closed and
strongly transitive subgroup of $\mathrm{Aut}_{0}(\Delta)$ is the group
$G^{+}$ associated with an isotropic simple algebraic group
$\operatorname{\mathbb{G}}$ over a non-Archimedean local field $k$.

The third result of the article concerns the Howe–Moore property [HM79] and we
generalize the proof given for the case of bi-regular trees [BM00b] to any
locally finite thick affine building $\Delta$. By Theorem 1.7, for an
irreducible $\Delta$, we do not obtain new examples of groups having the
Howe–Moore property. Still, the proof given in this article is different than
the strategy used so far in the literature (see for example [Cio]) and does
not relay on the polar decomposition $KA^{+}K$, where $K$ is a maximal compact
subgroup, and the important fact that $A^{+}$ is an abelian maximal sub-semi-
group.

The Howe–Moore property [HM79] is a harmonic analytic feature of a locally
compact group that has far-reaching applications to many areas of Mathematics,
such as homogeneous dynamics, geometry, or number theory. To list some, it has
notable consequences regarding discrete subgroups of semi-simple Lie groups,
Mostow’s rigidity theorem, Ratner’s theorems on unipotent flows, or mixing and
equidistribution results.

So far, all known examples of locally compact groups having the Howe–Moore
property are:

  1. 1.

connected, non-compact, simple real Lie groups, with finite center [HM79],
e.g., $\operatorname{SL}(n,\mathbb{R})$

  2. 2.

isotropic simple algebraic groups over non Archimedean local fields [HM79],
e.g., $\operatorname{SL}(n,\mathbb{Q}_{p})$

  3. 3.

closed, topologically simple subgroups of $\mathrm{Aut}(T)$ with a
$2$-transitive action on the boundary of a bi-regular tree $T$, that has
valence $\geq 3$ at every vertex, [BM00b], e.g., the universal group
$U(F)^{+}$ of Burger–Mozes, when F is $2$-transitive.

An important fact the above mentioned groups have in common is their polar
decomposition $KA^{+}K$, where $K$ is a maximal compact subgroup and $A^{+}$
is an abelian maximal sub-semi-group, the latter being crucial in the proof to
imply the Howe–Moore property. This was employed in [Cio] to give a unified
proof for all these above examples.

For a historical introduction and basic definitions regarding the Howe–Moore
property one can consult [Cio, Introduction and Section 2]. For completeness,
we recall here just some of the very basic notions.

Let $G$ be a locally compact group and $(\pi,\mathcal{H})$ a (strongly
continuous) unitary representation $\pi:G\to\mathcal{U}(\mathcal{H})$ on a
(infinite dimensional) complex Hilbert space
$(\mathcal{H},\left\langle\cdot,\cdot\right\rangle)$. For $v,w\in\mathcal{H}$
the associated $(v,w)$-matrix coefficient is the map $c_{v,w}:G\to\mathbb{C}$
given by $c_{v,w}(g):=\left\langle\pi(g)v,w\right\rangle$. We say $c_{v,w}$
vanishes at infinity if, for every $\epsilon>0$, the subset $\\{g\in
G\;|\;|c_{v,w}(g)|\geq\epsilon\\}$ is compact in $G$; equivalently,
$\lim\limits_{g\to\infty}c_{v,w}(g)=0$, where $\infty$ represents the one-
point compactification of the locally compact group $G$.

######  Definition 1.8.

A locally compact group $G$ has the Howe–Moore property if, any unitary
representation of $G$ that is without non-zero $G$–invariant vectors is
$C_{0}$ (i.e. all its matrix coefficients vanish at infinity).

######  Theorem 1.9.

Let $\Delta$ be a locally finite thick affine building and $G$ a topologically
simple, closed, strongly transitive and type-preserving subgroup of
$\mathrm{Aut}(\Delta)$. Then $G$ has the Howe–Moore property.

The two key ideas behind the proof of Theorem 1.9 are the use of the well
known Bruhat decomposition of a strongly transitive group
$G\leq\mathrm{Aut}(\Delta)$ (see Section 4), and the closed subgroup
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ introduced in
Definition 4.1. These allow us to get rid of the so-far-used polar
decomposition $KA^{+}K$ and its abelian sub-semi-group $A^{+}$ techniques.

In the case of a locally finite thick affine building $\Delta$ we expect the
Howe–Moore property of a closed, topologically simple and type-preserving
$G\leq\mathrm{Aut}(\Delta)$ to be equivalent to the strongly transitive action
of $G$ on $\Delta$. Recall, in the case of bi-regular trees, strong
transitivity is equivalent to $2$-transitivity. Moreover, it appears that
among locally finite thick buildings (so of any Coxeter type), only the affine
case provides examples of groups enjoying the Howe–Moore property. Both of the
above mentioned expectations will be studied in a future research project.

### Acknowledgements

Firstly, I would like to thank Tobias Hartnick for pushing me to prove
Theorems 1.5 and 1.7 during our research visit at the Institut des Hautes
Études Scientifiques (IHES) in Bures-sur-Yvette, Paris, and for very
illuminating conversations. As well, I am thankful to Stefan Witzel for
pointing out the articles [CM, CM1] and for very useful discussions. The first
part of the paper was written down and finalised during my research visit at
IHES and at the Institute of Mathematics of the Romanian Academy (IMAR),
Bucharest. I would like to thank those two institutions for the perfect
working conditions they provide.

##  2 From strong transitivity to Moufang’s condition

Let us recall a fundamental Rigidity Theorem of Tits. To state this, let $X$
be a building and $c$ a chamber of $X$. We denote by $E_{1}(c)$ the set of all
chambers of $X$ that are adjacent to $c$. In what follows we denote by
$\Delta$ an irreducible locally finite thick affine building of dimension
$\geq 2$, if not stated otherwise.

######  Theorem 2.1.

(see [AB, Theorem 5.205]) Let $X$ be a thick spherical building, and let $c$
and $c^{\prime}$ be opposite chambers in $X$. If an automorphism $\phi$ of $X$
fixes $E_{1}(c)\cup\\{c^{\prime}\\}$ pointwise, then $\phi$ is the identity.

From Theorem 2.1 one can deduce simple-transitivity for a root group
$U_{\alpha}$ when it enjoys the Moufang property. More precisely:

######  Corollary 2.2.

(see [AB, Lemma 7.25(3)]) Let $H$ be a closed subgroup of
$\mathrm{Aut}_{0}(\Delta)$ and $\alpha$ any root of $\partial_{\infty}\Delta$.
Suppose the root group $U_{\alpha}(H)$ acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$. Then the action of
$U_{\alpha}(H)$ on $\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$ is simple-
transitive; i.e., for every $u\in U_{\alpha}(H)-\\{id\\}$ the action of $u$
has no fixed points on $\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$.

###### Proof.

Suppose there is an element $u\in U_{\alpha}(H)-\\{id\\}$ that fixes an
element of $\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$, i.e., $u$ fixes
pointwise at least one apartment $\Sigma$ of $\partial_{\infty}\Delta$ that
contains $\alpha$ as a half-apartment. As the affine building $\Delta$ is
irreducible, its Coxeter diagram has no isolated nodes, and thus also the one
of the spherical building $\partial_{\infty}\Delta$. Then by [AB, Example
3.128] there is an ideal chamber $c\in\mathrm{Ch}(\alpha)$ having no panel in
$\partial\alpha$. Then by the definition of $U_{\alpha}(H)$, the element $u$
fixes $E_{1}(c)\subset\partial_{\infty}\Delta$ and the apartment $\Sigma$, and
thus the ideal chamber $c^{\prime}$ of $\Sigma$ opposite $c$. By Theorem 2.1,
as $u$ is also an automorphism of $\partial_{\infty}\Delta$, we must have
$u|_{\partial_{\infty}\Delta}=\operatorname{id}$, given a contradiction with
our assumption. ∎

As written in [AB, Remark 7.26], the assumption that the Coxeter diagram has
no isolated nodes (e.g., the irreducibility condition from Corollary 2.2) is
crucial and cannot be removed.

######  Corollary 2.3.

Let $\alpha$ be any root of $\partial_{\infty}\Delta$ and $H$ a closed
subgroup of $\mathrm{Aut}_{0}(\Delta)$. Suppose both root groups
$U_{\alpha}(H)$ and $U_{\alpha}(\mathrm{Aut}_{0}(\Delta))$ act transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$. Then
$U_{\alpha}(\mathrm{Aut}_{0}(\Delta))=U_{\alpha}(H)$.

###### Proof.

It is obvious that $U_{\alpha}(H)\leq U_{\alpha}(\mathrm{Aut}_{0}(\Delta))$.
By Corollary 2.2 we have that both groups
$U_{\alpha}(H),U_{\alpha}(\mathrm{Aut}_{0}(\Delta))$ act simple-transitively
on $\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$ and so they must be equal.
∎

######  Lemma 2.4.

Let $\Delta$ be a locally finite thick affine building. Let $H$ be a closed
subgroup of $\mathrm{Aut}_{0}(\Delta)$ that acts strongly transitively on
$\Delta$. Then for every $\xi\in\partial_{\infty}\Delta$ the group $H_{\xi}$
acts cocompactly on $\Delta$ and transitively on the set
$\mathrm{Opp}(\xi)\subset\partial_{\infty}\Delta$ of all ideal points opposite
$\xi$.

###### Proof.

Fix an apartment $\Sigma$ in $\Delta$. By the strong transitivity of $H$ on
$\Delta$, it is enough to consider only ideal points
$\xi\in\partial_{\infty}\Sigma$. For what follows, fix such $\xi$. Then there
is a unique, minimal-dimensional ideal simplex
$\sigma(\xi)\subset\partial_{\infty}\Sigma$ such that $\xi$ is an interior
point of $\sigma(\xi)$. Then choose an ideal chamber $c\in\mathrm{Ch}(\Sigma)$
(not necessarily unique), with $\sigma(\xi)\subset c$ and notice $H_{c}\leq
H_{\xi}$. It is then enough to prove that for every $c\in\mathrm{Ch}(\Sigma)$,
the group $H_{c}$ acts cocompactly on $\Delta$. Indeed, fix
$c\in\mathrm{Ch}(\Sigma)$. Since $H$ is strongly transitive on $\Delta$, by
[Ga, Sec. 17.1], $H$ is strongly transitive on $\partial_{\infty}\Delta$ as
well. In particular, we have that $H_{c}$ acts transitively on the set
$\mathrm{Opp}(c)$ of all ideal chambers of $\partial_{\infty}\Delta$ opposite
$c$. As for every chamber $C\in\mathrm{Ch}(\Delta)$ there is an apartment
$\Sigma_{1}$ in $\Delta$ containing $C$, and $c$ as an ideal chamber of
$\mathrm{Ch}(\Sigma_{1})$, we are left with proving that
$\mathrm{Stab}_{H_{c}}(\Sigma):=\\{g\in H_{c}\;|\;g(\Sigma)=\Sigma\\}$ acts
cocompactly on $\Sigma$. To do that, again by the strong transitivity of $H$
on $\Delta$, we know that
$\mathrm{Stab}_{H}(\Sigma)\backslash\mathrm{Fix}_{H}(\Sigma)$ is the affine
Weyl group $W$ corresponding to the affine building $\Delta$, and that W is
generated by the reflexions through the walls of a chamber in
$\mathrm{Ch}(\Sigma)$ (see [Ga, Section 5.2]). We also know that $W$ is simply
transitive on special vertices of $\Sigma$ of the same type, and for every two
special vertices $v_{1},v_{2}$ of $\Sigma$ of the same type there is a
translation automorphism in $\mathrm{Stab}_{H}(\Sigma)$ of the apartment
$\Sigma$, sending $v_{1}$ to $v_{2}$. In particular, such translation
automorphism fixes pointwise the ideal boundary $\partial_{\infty}\Sigma$ of
$\Sigma$, and thus belonging to the group $\mathrm{Stab}_{H_{c}}(\Sigma)\leq
H_{c}$. This remark easily implies $\mathrm{Stab}_{H_{c}}(\Sigma)$ acts
cocompactly on $\Sigma$ and so proving the claim.

Let us now prove the transitivity of $H_{\xi}$ on the set $\mathrm{Opp}(\xi)$.
As above, fix an apartment $\Sigma$ of $\Delta$ with
$\xi\in\partial_{\infty}\Sigma$, an ideal chamber
$c\in\mathrm{Ch}(\partial_{\infty}\Sigma)$ with $\xi\in c$, and let $\xi_{-}$
be the unique ideal point in $\partial_{\infty}\Sigma$ opposite $\xi$. Take
$\xi^{\prime}\in\mathrm{Opp}(\xi)$, with $\xi^{\prime}\neq\xi_{-}$. Then there
is an apartment $\Sigma^{\prime}$ of $\Delta$ such that
$c\subset\mathrm{Ch}(\partial_{\infty}\Sigma^{\prime})$ and
$\xi^{\prime}\in\partial_{\infty}\Sigma^{\prime}$. As $H_{c}\leq H_{\xi}$ acts
transitively on $\mathrm{Opp}(c)$, there is $g\in H_{c}$ with
$g(\Sigma^{\prime})=\Sigma$ and $g(c)=c$. In particular,
$g(\xi^{\prime})\in\partial_{\infty}\Sigma$ and $g(\xi)=\xi$. As
$\xi_{-},\xi^{\prime}$ are opposite $\xi$, we must have
$g(\xi^{\prime})=\xi_{-}$. The lemma is proven. ∎

Let $\xi\in\partial_{\infty}\Delta$ and choose
$\xi_{-}\in\partial_{\infty}\Delta$ an ideal point opposite $\xi$. Denote by
$P(\xi,\xi_{-})$ the set of all apartments of $\Delta$ that contain $\xi$ and
$\xi_{-}$ in their ideal boundary at infinity. As $\xi$ is an ideal point,
there is a unique minimal-dimensional ideal simplex
$\sigma(\xi)\subset\partial_{\infty}\Delta$ such that $\xi$ is an interior
point of $\sigma(\xi)$. Let $\sigma_{-}(\xi)\subset\partial_{\infty}\Delta$ be
the corresponding ideal simplex, opposite $\sigma(\xi)$, that contains
$\xi_{-}$ as an interior point. Denote by $n_{\xi}$ the dimension (with
respect to $\partial_{\infty}\Delta$) of the ideal simplex $\sigma(\xi)$. Then
$P(\xi,\xi_{-})=\mathbb{R}^{n_{\xi}+1}\times\Delta_{\xi}$, where
$\Delta_{\xi}$ is a sub-building of $\Delta$, this is proved by Rousseau [Rou,
4.3] in the more general setting of masures.

######  Lemma 2.5.

Let $\Delta$ be a locally finite thick affine building. Let $H$ be a closed
subgroup of $\mathrm{Aut}_{0}(\Delta)$ that acts strongly transitively on
$\Delta$. Let $\xi\in\partial_{\infty}\Delta$ and choose
$\xi_{-}\in\partial_{\infty}\Delta$ opposite $\xi$. Then the closed group
$H_{\xi,\xi_{-}}\leq H_{\xi}$ acts cocompactly on $P(\xi,\xi_{-})$.

###### Proof.

Fix an apartment $\Sigma$ of $\Delta$ such that
$\xi,\xi_{-}\in\partial_{\infty}\Sigma$, and an ideal chamber
$c\in\mathrm{Ch}(\partial_{\infty}\Sigma)$ with $\xi\in c$. Then $\Sigma\in
P(\xi,\xi_{-})$. Take another apartment $\Sigma^{\prime}\in P(\xi,\xi_{-})$
with $\Sigma\neq\Sigma^{\prime}$, and an ideal chamber
$c^{\prime}\in\mathrm{Ch}(\partial_{\infty}\Sigma^{\prime})$ with $\xi\in
c^{\prime}$. By [Ga, Sec. 17.1], $H$ acts strongly transitively on
$\partial_{\infty}\Delta$. Then there is $g\in H$ such that $g(c^{\prime})=c$
and $g(\Sigma^{\prime})=\Sigma$. Since $\Sigma,\Sigma^{\prime}\in
P(\xi,\xi_{-})$, we must have $g(\xi)=\xi$ and $g(\xi_{-})=\xi_{-}$.
Therefore, $g\in H_{\xi,\xi_{-}}$ and this shows $H_{\xi,\xi_{-}}$ acts
transitively on the set of apartments $P(\xi,\xi_{-})$. To prove the lemma it
is enough to show that $\mathrm{Stab}_{H_{\xi,\xi_{-}}}(\Sigma)$ acts
cocompactly on $\Sigma$. As used in the proof of Lemma 2.4, the affine Weyl
group $W$ of $\Delta$, which is isomorphic to
$\mathrm{Stab}_{H}(\Sigma)\backslash\mathrm{Fix}_{H}(\Sigma)$, contains
translation automorphisms in $\mathrm{Stab}_{H}(\Sigma)$ that sends any
special vertex of $\Sigma$ to any other special vertex of $\Sigma$ of the same
type. And all those translation automorphisms in $\mathrm{Stab}_{H}(\Sigma)$
do pointwise stabilise the ideal boundary $\partial_{\infty}\Sigma$. Thus,
they are all elements of $H_{\xi,\xi_{-}}$, and so
$\mathrm{Stab}_{H_{\xi,\xi_{-}}}(\Sigma)$ indeed acts cocompactly on $\Sigma$
as claimed, implying the lemma. ∎

Let us now recall some definitions and results from [CM].

######  Definition 2.6 (see Introduction [CM]).

Let $(X,d)$ be a proper $\mathrm{CAT}(0)$ space, $H\leq Isom(X)$ a closed
subgroup and $\xi$ in the visual boundary $\partial_{\infty}X$ of $X$.
Associated with $H_{\xi}$ we define

$H_{\xi}^{u}:=\\{g\in
H\;|\;\lim\limits_{t\to\infty}d(g(r(t)),r(t))=0,\;\forall\text{ geodesic rays
}r:[0,\infty)\to X\text{ with }r(\infty)=\xi\\}.$

Notice, $H_{\xi}^{u}$ is a closed normal subgroup of $H_{\xi}$.

######  Remark 2.7.

Let $\Delta$ be a locally finite thick affine building and $H$ a closed
subgroup of $\mathrm{Aut}_{0}(\Delta)$. Then it is easy to see directly from
Definition 2.6 that for every $\xi\in\partial_{\infty}\Delta$ the subgroup
$H_{\xi}^{u}$ fixes pointwise any ideal chamber of $\partial_{\infty}\Delta$
that contains $\xi$.

Below we state [CM, Theorem J] in the spacial case of $\Delta$ being a locally
finite thick affine building.

######  Theorem 2.8.

(Levi decomposition I) Let $\Delta$ be a locally finite thick affine building
and $H$ a closed subgroup of $\mathrm{Aut}_{0}(\Delta)$ that acts strongly
transitively on $\Delta$. Let $\xi\in\partial_{\infty}\Delta$. Then for every
$\xi^{\prime}\in\mathrm{Opp}(\xi)$ we have a decomposition

$H_{\xi}=H_{\xi,\xi^{\prime}}\cdot H_{\xi}^{u}$

which is almost semi-direct in the sense that $H_{\xi,\xi^{\prime}}\cap
H_{\xi}^{u}$ is compact. In particular, $H_{\xi}^{u}$ acts transitively on
$\mathrm{Opp}(\xi)$.

###### Proof.

For the complete proof see [CM]. Here we will just give the main idea of the
proof. Fix for what follows some $\xi^{\prime}\in\mathrm{Opp}(\xi)$ and
consider the set $P(\xi,\xi^{\prime})$. By Lemma 2.4 we know that $H_{\xi}$
acts transitively on $\mathrm{Opp}(\xi)$, implying that for every
$\xi^{\prime\prime}\in\mathrm{Opp}(\xi)$, there exists $g\in H_{\xi}$ such
that $g(\xi^{\prime\prime})=\xi^{\prime}$ and in particular,
$g(P(\xi,\xi^{\prime\prime}))=P(\xi,\xi^{\prime})$. Moreover, similarly as for
retractions with respect to an ideal chamber in $\partial_{\infty}\Delta$, one
can intuitively consider a canonical retraction $\rho_{\xi}$ with respect to
the ideal point $\xi$, sending each set $P(\xi,\xi^{\prime\prime})$ to
$P(\xi,\xi^{\prime})$, for every $\xi^{\prime\prime}\in\mathrm{Opp}(\xi)$. The
horoaction $\omega_{\xi}:H_{\xi}\to Isom(P(\xi,\xi^{\prime}))$ defined in [CM,
Definition 3.1] is aiming to track the action of an element $g\in H_{\xi}$,
via the retraction $\rho_{\xi}$ on the space $P(\xi,\xi^{\prime})$ (i.e.,
$\rho_{\xi}(g)$). Then it is easy to see that
$\mathrm{Ker}(\omega_{\xi})=H_{\xi}^{u}$ and by the transitivity of $H_{\xi}$
on $\mathrm{Opp}(\xi)$, $\mathrm{Im}(\omega_{\xi})=H_{\xi,\xi^{\prime}}$. The
decomposition $H_{\xi}=H_{\xi,\xi^{\prime}}\cdot H_{\xi}^{u}$ follows. As
$H_{\xi}^{u}$ contains only elliptic elements, the intersection
$H_{\xi,\xi^{\prime}}\cap H_{\xi}^{u}$ is compact. Again by the transitivity
of $H_{\xi}$ on $\mathrm{Opp}(\xi)$ one can conclude that $H_{\xi}^{u}$ acts
transitively on $\mathrm{Opp}(\xi)$ as well. ∎

######  Definition 2.9 (see [CM] Definition 3.5).

Let $G$ be a group acting by isometries on a proper $\mathrm{CAT}(0)$ space
$X$. A sequence $\\{g_{n}\\}_{n\geq 1}\subset G$ is called radial in $G$ if
there exist an ideal point $\xi\in\partial_{\infty}X$, called the radial limit
point of $\\{g_{n}\\}_{n\geq 1}$, and a geodesic ray $r:[0,\infty)\to X$
pointing towards $\xi$ such that the sequence $g_{n}^{-1}(r(0))$ converges to
$\xi$, with respect to the cone topology on $X\cup\partial_{\infty}X$, while
remaining at bounded distance from $r$.

One of the properties of radial sequences is the existence of a limiting bi-
infinite geodesic line in $X$ with specific features. The following technical
lemma expresses exactly that, although its proof is very easy and relays on
the properness of $X$ implying that $X\cup\partial_{\infty}X$ is a compact
space, with respect to the cone topology.

######  Lemma 2.10.

Let $G$ be a group acting by isometries on a proper $\mathrm{CAT}(0)$ space
$X$. Suppose $\\{g_{n}\\}_{n\geq 1}\subset G$ is a radial sequence with radial
limit point $\xi\in\partial_{\infty}X$ and geodesic ray $r:[0,\infty)\to X$
pointing towards $\xi$. Then one can extract a subsequence
$\\{g_{n_{k}}\\}_{k\geq 1}$ of $\\{g_{n}\\}_{n\geq 1}$ such that the sequence
of geodesic rays $\\{g_{n_{k}}(r)\\}_{k\geq 1}$ converges to a bi-infinite
geodesic line $\ell:\mathbb{R}\to X$ with the properties that
$\lim\limits_{k\to\infty}g_{n_{k}}(r(0))=\ell(-\infty)=:\eta_{-}\in\partial_{\infty}X$
and
$\lim\limits_{k\to\infty}g_{n_{k}}(\xi)=\ell(\infty)=:\eta_{+}\in\partial_{\infty}X$.

###### Proof.

By the definition of a radial sequence, we have that
$\\{g_{n}^{-1}(r(0))\\}_{n\geq 1}$ converges to $\xi$ and it stays at bounded
distance from $r$. Denote by $r(t_{n})$ the projection of $g_{n}^{-1}(r(0))$
to $r$. Then, by the construction, the sequence $\\{g_{n}(r(t_{n}))\\}_{n\geq
1}$ stays at bounded distance from $r(0)$. As $X$ is proper, implying the
visual boundary $\partial_{\infty}X$ is compact, one can extract a subsequence
$\\{n_{k}\\}_{k\geq 1}$ such that
$\lim\limits_{k\to\infty}g_{n_{k}}(\xi)=\eta_{+}\in\partial_{\infty}X$. Then,
up to further extracting a subsequence from $\\{n_{k}\\}_{k\geq 1}$, the
sequence of points $\\{g_{n_{k}}(r(0))\\}_{k\geq 1}$ converges to an ideal
point $\eta_{-}\in\partial_{\infty}X$. As $g_{n_{k}}(r)$ are geodesics rays,
from the above limiting procedures one can conclude that the sequence of
geodesic rays $\\{g_{n_{k}}(r)\\}_{k\geq 1}$ converges to a bi-infinite
geodesic line $\ell:\mathbb{R}\to X$ with $\ell(-\infty)=\eta_{-}$ and
$\ell(\infty)=\eta_{+}$. ∎

######  Definition 2.11.

In the hypotheses of Lemma 2.10, let $\\{g_{n}\\}_{n\geq 1}\subset G$ be a
radial sequence with radial limit point $\xi\in\partial_{\infty}X$ and
geodesic ray $r:[0,\infty)\to X$ pointing towards $\xi$. An ideal point
$\eta\in\partial_{\infty}X$ such that there is a subsequence
$\\{n_{k}\\}_{k\geq 1}$ with
$\lim\limits_{k\to\infty}g_{n_{k}}(r(0))=\eta\in\partial_{\infty}X$ is called
a limit point of $\\{g_{n}\\}_{n\geq 1}$.

The following, also technical lemma, gives a reason why and how radial
sequences can be used in applications.

######  Lemma 2.12.

Let $X$ be a proper $\mathrm{CAT}(0)$ space and $G\leq Isom(X)$ be a closed
group. Suppose $\\{g_{n}\\}_{n\geq 1}\subset G$ is a radial sequence with
radial limit point $\xi\in\partial_{\infty}X$ and geodesic ray
$r:[0,\infty)\to X$ pointing towards $\xi$. Suppose also the sequence of
geodesic rays $\\{g_{n}(r)\\}_{n\geq 1}$ converges to a bi-infinite geodesic
line $\ell:\mathbb{R}\to X$ with the properties that
$\lim\limits_{n\to\infty}g_{n}(r(0))=\ell(-\infty)=:\eta_{-}\in\partial_{\infty}X$
and
$\lim\limits_{n\to\infty}g_{n}(\xi)=\ell(\infty)=:\eta_{+}\in\partial_{\infty}X$.
Let $g\in G_{\xi}$. Then the sequence $\\{g_{n}gg_{n}^{-1}\\}_{n\geq 1}$ is
bounded and for any of its convergent subsequence
$\\{g_{n_{k}}gg_{n_{k}}^{-1}\\}_{n\geq 1}$ the corresponding limit
$\lim\limits_{k\to\infty}g_{n_{k}}gg_{n_{k}}^{-1}=:h\in G$ is an element of
$G_{\eta_{-},\eta_{+}}$.

###### Proof.

As $g\in G_{\xi}$ and by the definition of the visual boundary
$\partial_{\infty}X$, we first notice that the geodesic ray $g(r)$ is
asymptotically equivalent to $r$, and so at bounded distance from the geodesic
ray $r$. As $\\{g_{n}^{-1}(r(0))\\}_{n\geq 1}$ remains at bounded distance
from $r$, we have that indeed
$d(g_{n}gg_{n}^{-1}(r(0)),r(0))=d(gg_{n}^{-1}(r(0)),g_{n}^{-1}(r(0)))$ is a
bounded sequence. Take now a convergent subsequent
$\lim\limits_{k\to\infty}g_{n_{k}}gg_{n_{k}}^{-1}=:h\in G$. By the hypothesis
$\lim\limits_{n\to\infty}g_{n}(\xi)=\eta_{+}$ and by the continuity of the
action of $G$ on $\partial_{\infty}X$, we also have
$\lim\limits_{k\to\infty}g_{n_{k}}gg_{n_{k}}^{-1}(g_{n_{k}}(\xi))=\lim\limits_{k\to\infty}g_{n_{k}}g(\xi)=\lim\limits_{k\to\infty}g_{n_{k}}(\xi)=\eta_{+}=\lim\limits_{k\to\infty}h(g_{n_{k}}(\xi))=h(\eta_{+})$.
Now using $\lim\limits_{n\to\infty}g_{n}(r(0))=\eta_{-}$ and the same
computation as above, we have
$\lim\limits_{k\to\infty}g_{n_{k}}gg_{n_{k}}^{-1}(g_{n_{k}}(r(0)))=\lim\limits_{k\to\infty}g_{n_{k}}g(r(0))=\eta_{-}=\lim\limits_{k\to\infty}h(g_{n_{k}}(r(0)))=h(\eta_{-})$.

Notice, as each element $g_{n}gg_{n}^{-1}$ inherits the same properties as the
element $g$ (i.e., elliptic if $g$ is elliptic, hyperbolic of the same
translation length if $g$ is hyperbolic), the same is true for a limiting
element $h$. ∎

######  Lemma 2.13.

Let $\Delta$ be a locally finite thick affine building. Let $H$ be a closed
subgroup of $\mathrm{Aut}_{0}(\Delta)$ that acts strongly transitively on
$\Delta$. Then every $\xi\in\partial_{\infty}\Delta$ is a radial limit point
for some radial sequence in $H_{\xi}$.

###### Proof.

Let $\xi\in\partial_{\infty}\Delta$ and choose an apartment $\Sigma$ of
$\Delta$ such that $\xi\in\partial_{\infty}\Sigma$. Let
$r:[0,\infty)\to\Sigma$ be a geodesic ray with endpoint $\xi$ and $r(0)$ a
special vertex of $\Sigma$. Notice, there are an infinite number of special
vertices of the same type as $r(0)$ that are at bounded distance from $r$, and
converging to the endpoint $\xi$. By using the affine Weil group $W$ of
$\Delta$ as in the proof of Lemma 2.4, for each special vertex $v$ of
$\Sigma$, of the same type as $r(0)$, there is a translation automorphism
$g_{v}\in\mathrm{Stab}_{H}(\Sigma)$ sending $r(0)$ to $v$. By choosing a
sequence of special vertices $\\{v_{n}\\}_{n\geq 0}$, of the same type as
$r(0)$, at bounded distance from $r$, and converging to $\xi$, (the inverses
of) their corresponding translation automorphisms $\\{g_{v_{n}}\\}_{n\geq
1}\in\mathrm{Stab}_{H}(\Sigma)$ will do the job. ∎

We are now ready to state the Levi decomposition for subgroups $N\leq
H_{\xi}$, where $H\leq\mathrm{Aut}(\Delta)$ and
$\xi\in\partial_{\infty}\Delta$. This is [CM, Theorem 3.12] specialised to the
case of an affine building $\Delta$.

######  Theorem 2.14.

(Levi decomposition II) Let $\Delta$ be a locally finite thick affine building
and $H$ a closed subgroup of $\mathrm{Aut}_{0}(\Delta)$ that acts strongly
transitively on $\Delta$. Let $\xi\in\partial_{\infty}\Delta$. Let $N\leq
H_{\xi}$ be a closed subgroup that is normalised by some radial sequence
$\\{g_{n}\\}_{n\geq 1}\subset H_{\xi}$ with radial limit point $\xi$ and
unique limit point $\xi_{-}$; hence $\xi_{-}\in\mathrm{Opp}(\xi)$. Then,
writing $N^{u}:=N\cap H_{\xi}^{u}$, we have the decomposition

$N=N_{\xi_{-}}\cdot N^{u}$

which is almost semi-direct in the sense that $N_{\xi_{-}}\cap N^{u}$ is
compact. In particular, $N^{u}$ acts transitively on the $N$-orbits of
$\xi_{-}$ in $\mathrm{Opp}(\xi)$.

###### Proof.

For the proof see [CM, Theorem 3.12]. Here we only give the main idea and the
intuition behind the proof. As the radial sequence $\\{g_{n}\\}_{n\geq 1}$ has
the property that belongs to $H_{\xi}$ we have
$\lim\limits_{n\to\infty}g_{n}(r(0))=\xi_{-}\in\partial_{\infty}X$ and
$\lim\limits_{n\to\infty}g_{n}(\xi)=\xi$. Then for every $g\in N$, by Lemma
2.12 applied to $\\{g_{n}\\}_{n\geq 1}$, any of the convergent subsequence
$\\{g_{n_{k}}gg_{n_{k}}^{-1}\\}_{n\geq 1}$ has the corresponding limit
$\lim\limits_{k\to\infty}g_{n_{k}}gg_{n_{k}}^{-1}=h$ in $G_{\xi_{-},\xi}$. As
$N$ is normalised by the radial sequence $\\{g_{n}\\}_{n\geq 1}$, we have
moreover that $\lim\limits_{k\to\infty}g_{n_{k}}gg_{n_{k}}^{-1}=h\in
N_{\xi_{-}}$. Because the properties of $g$ are preserved by each of the
elements $g_{n_{k}}gg_{n_{k}}$, then also the limit $h$ inherits the
properties of $g$ (i.e., elliptic if $g$ is elliptic, hyperbolic of the same
translation length if $g$ is hyperbolic). This intuitively implies that the
horoaction $\omega_{\xi}:H_{\xi}\to Isom(P(\xi,\xi{-}))$, defined in [CM,
Definition 3.1], restricted to $N$ has the property that
$\omega_{\xi}(N)=\omega_{\xi}(N_{\xi_{-}})$. This will then give the
decomposition $N=N_{\xi_{-}}\cdot N^{u}$, where $N^{u}:=N\cap
H_{\xi}^{u}=N\cap\mathrm{Ker}(\omega_{\xi})$. ∎

###### Proof of Theorem 1.5.

The proof is the same as in [CM1, Section 7.G]. Still, because we work with
subgroups $H\leq\mathrm{Aut}_{0}(\Delta)$, and not only with
$\mathrm{Aut}_{0}(\Delta)$, we need to treat the general case of $\Delta$
being of any dimension $\geq 2$. In [CM1, Section 7.G], because they work with
$\mathrm{Aut}_{0}(\Delta)$, they can reduce directly to the dimension of
$\Delta$ being $2$ by using the Theorem of Tits that every irreducible,
locally finite thick affine building of dimension $\geq 3$ is Moufang (see
[AB, Chapter 7]).

Let $\alpha$ be a root of $\partial_{\infty}\Delta$ and for what follows fix
an apartment $\Sigma$ in $\Delta$ such that
$\alpha\subset\partial_{\infty}\Sigma$. Then the pointwise stabiliser in $H$
of the ideal half-apartment $\alpha$ is denoted by $H_{\alpha}:=\\{g\in
H\;|\;g(\alpha)=\alpha\text{ pointwise}\\}$. Then it is easy to see that
$H_{\alpha}$ is a closed subgroup of $H$. Because $H$ is strongly transitive
on $\Delta$ implying $H$ is strongly transitively on $\partial_{\infty}\Delta$
by [Ga, Sec. 17.1], one has $H_{\alpha}$ acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$, the set of all apartments of
$\partial_{\infty}\Delta$ that contain $\alpha$ as a half-apartment. We claim
$H_{\alpha}\cap U_{\alpha}(H)$ contains a subgroup that is still transitive on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$, where $U_{\alpha}(H)$ is the
root group in $H$ associated with $\alpha$. This claim will definitely imply
that $H$ is Moufang.

It remains to prove the claim. Indeed, as $\partial_{\infty}\Delta$ is a
spherical building, the number of chambers of $\alpha$ is finite, and
implicitly the number of panels $P$ in $\alpha-\partial\alpha$ is also finite,
say $N\geq 1$. We enumerate them by $P_{1},P_{2},...,P_{N}$ and for each
$i\in\\{1,...,N\\}$, we choose an ideal point
$\xi_{i}\in\partial_{\infty}\Delta$ that is in the interior of the panel
$P_{i}$. By Remark 2.7, for every $i\in\\{1,...,N\\}$, the closed subgroup
$H_{\xi_{i}}^{u}$ fixes pointwise any chamber of $\partial_{\infty}\Delta$
that contains $\xi_{i}$, and equivalently that contains $P_{i}$ as a panel.
Thus, for every $i\in\\{1,...,N\\}$, the group $H_{\xi_{i}}^{u}$ fixes
pointwise $\mathrm{St}_{\partial_{\infty}\Delta}(P_{i})$. By Lemma 2.13, each
$\xi_{i}$, for $i\in\\{1,...,N\\}$, is a radial limit point and we denote by
$\\{t_{i,n}\\}_{n\geq 1}$ an associated radial sequence in $H_{\xi_{i}}$. By
the proof of Lemma 2.13 one can choose $\\{t_{i,n}\\}_{n\geq
1}\subset\mathrm{Stab}_{H}(\Sigma)$ in such a way that they are moreover
translation automorphisms of $\Sigma$, for each $i\in\\{1,...,N\\}$. In
particular, $\\{t_{i,n}\\}_{n\geq 1}$ normalises $H_{\xi_{i}}^{u}$, fixes
pointwise the ideal boundary $\partial_{\infty}\Sigma$, thus normalises
$H_{\alpha}$ as well. Notice,
$H_{\alpha}\leq\bigcap\limits_{i=1}^{N}H_{\xi_{i}}$.

We denote $H_{\alpha}=:V_{0}$, and for every $i\in\\{1,...,N\\}$ we put,
recursively, $V_{i}:=V_{i-1}\cap H_{\xi_{i}}^{u}$. We have that $V_{0}\geq
V_{1}\geq V_{2}\geq...\geq V_{N}$ and they are all closed subgroups of $H$.
Then, by construction we have $V_{N}$ is a subgroup of $U_{\alpha}(H)$. We
will prove by induction on $i\in\\{1,...,N\\}$ that each of $V_{i}$ acts
transitively on $\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$, implying that
$V_{N}$ is a subgroup of $U_{\alpha}(H)$ that acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$.

Take $i=1$. Then the radial sequence $\\{t_{1,n}\\}_{n\geq 1}\subset
H_{\xi_{1}}$, with radial limit point $\xi_{1}$ normalises both
$H_{\alpha}=V_{0}$ and $H_{\xi_{1}}^{u}$, and so also the intersection
$V_{1}=V_{0}\cap H_{\xi_{1}}^{u}$. Then by Theorem 2.14 applied to $V_{0}$ (up
to extraction a subsequence) we have $V_{0}=V_{0,\xi_{-1}}\cdot V_{1}$, where
$\xi_{-1}\in\mathrm{Opp}(\xi_{1})\cap\partial_{\infty}\Sigma$, and $V_{1}$
acts transitively on the $V_{0}$-orbits of $\xi_{-1}$ in
$\mathrm{Opp}(\xi_{1})$. Since $V_{0}$ acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$, and since there is only one
apartment in $\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$ which contains
$\xi_{-1}$ and $\alpha$, one deduces $V_{1}$ acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$ as well.

For $2\leq i\leq N$ we do the same as above by replacing the index $1$ with
$i$, and the index $0$ by $i-1$: The radial sequence $\\{t_{i,n}\\}_{n\geq
1}\subset H_{\xi_{i}}$, with radial limit point $\xi_{i}$ normalises
$H_{\alpha}$ and $H_{\xi_{j}}^{u}$, for each $j\in\\{1,...,N\\}$, thus
$V_{i-1}$ (by induction step), and so also the intersection $V_{i}=V_{i-1}\cap
H_{\xi_{i}}^{u}$. Then by Theorem 2.14 applied to $V_{i-1}$ (up to extraction
a subsequence) we have $V_{i-1}=V_{i-1,\xi_{-i}}\cdot V_{i}$, where
$\xi_{-i}\in\mathrm{Opp}(\xi_{i})\cap\partial_{\infty}\Sigma$, and $V_{i}$
acts transitively on the $V_{i-1}$-orbits of $\xi_{-i}$ in
$\mathrm{Opp}(\xi_{i})$. Since $V_{i-1}$ acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$ (by the induction step), and
since there is only one apartment in
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$ which contains $\xi_{-i}$ and
$\alpha$, one deduces $V_{i}$ acts transitively on
$\mathcal{A}_{\partial_{\infty}\Delta}(\alpha)$ as well. The conclusion of the
theorem follows. ∎

##  3 Proof of Theorem 1.7

Let $\Delta$ be an irreducible, locally finite thick affine building of
dimension $\geq 2$ such that $\mathrm{Aut}_{0}(\Delta)$ is strongly transitive
on $\Delta$, thus also on $\partial_{\infty}\Delta$ by [Ga, Sec. 17.1]. For
what follows, the irreducibility cannot be removed as noticed after the
Corollary 2.2. By Theorem 1.5 $\mathrm{Aut}_{0}(\Delta)$ is Moufang, and so is
the spherical building $\partial_{\infty}\Delta$ (in the sense of [AB,
Definition 7.27]). Fix an apartment $\Sigma$ of $\partial_{\infty}\Delta$ and
an ideal chamber $c_{+}\in\mathrm{Ch}(\Sigma)$ with its opposite chamber
$c_{-}\in\mathrm{Ch}(\Sigma)$. Denote by $\Phi$ the set of all roots of
$\Sigma$ and by $\Phi^{+}$ the roots in $\Phi$ that contain $c$ as a chamber;
we call $\Phi^{+}$ the set of positive roots with respect to $c_{+}$. We set

$U^{+}:=\langle
U_{\alpha}(\mathrm{Aut}_{0}(\Delta))\;|\;\alpha\in\Phi^{+}\rangle\text{ and
}U^{-}:=\langle
U_{\alpha}(\mathrm{Aut}_{0}(\Delta))\;|\;-\alpha\in\Phi^{+}\rangle.$

######  Remark 3.1.

For any closed and strongly transitive subgroup $H$ of
$\mathrm{Aut}_{0}(\Delta)$, Theorem 1.5 and Corollary 2.3 tell us that we
obtain the same groups $U^{+}$, resp., $U^{-}$ if we replace
$U_{\alpha}(\mathrm{Aut}_{0}(\Delta))$ by $U_{\alpha}(H)$.

Because we work with the spherical building $\partial_{\infty}\Delta$ which is
Moufang, we can apply the results of [Ro, Chapter 6 and Thm. 6.18]. We recall
the notation. Let

$B^{+}:=\mathrm{Stab}_{\mathrm{Aut}_{0}(\Delta)}(c_{+})\;\text{ and
}\;B^{-}:=\mathrm{Stab}_{\mathrm{Aut}_{0}(\Delta)}(c_{-})$

be the positive, resp., negative, minimal parabolic subgroup of
$\mathrm{Aut}_{0}(\Delta)$.

For a closed subgroup $H\leq\mathrm{Aut}_{0}(\Delta)$ we set

$B(H)^{+}:=\mathrm{Stab}_{H}(c_{+})\leq
B^{+}\;\text{and}\;B(H)^{-}:=\mathrm{Stab}_{H}(c_{-})\leq B^{-}.$

######  Proposition 3.2.

Let $H$ be a closed and strongly transitive subgroup of
$\mathrm{Aut}_{0}(\Delta)$. Then $B^{\pm}=U^{\pm}\rtimes M$ and
$B(H)^{\pm}=U^{\pm}\rtimes M(H)$, where $M:=B^{+}\cap B^{-}$ and
$M(H):=B(H)^{+}\cap B(H)^{-}$. In particular, $U^{+}$ (resp., $U^{-}$) is a
normal subgroup in both subgroups $B^{+}$ and $B(H)^{+}$ (resp., $B^{-}$ and
$B(H)^{-}$).

###### Proof.

This follows directly from [Ro, Chapter 6, Thms. 6.17 and 6.18] and Corollary
2.3 above. ∎

######  Proposition 3.3.

Let $H$ be a closed and strongly transitive subgroup of
$\mathrm{Aut}_{0}(\Delta)$. Then $\overline{\langle U^{+},U^{-}\rangle}$ is
normal in $H$ and $\langle U^{+},U^{-}\rangle=H^{+}$.

###### Proof.

Let us prove first that $\overline{\langle U^{+},U^{-}\rangle}$ is normal in
$H$. Indeed, by [Cio, Prop. 4.11] we know $H=\langle
H_{c_{+}}^{0},H_{c_{-}}^{0}\rangle$, where
$H_{c_{\pm}}^{0}=(B(H)^{\pm})^{0}:=\\{g\in B(H)^{\pm}\;|\;g\text{
elliptic}\\}$. Notice $U^{\pm}\leq H_{c_{\pm}}^{0}$ and by Proposition 3.2 we
have $U^{\pm}$ is normal in $H_{c_{\pm}}^{0}$. To prove $\overline{\langle
U^{+},U^{-}\rangle}$ is normal in $H$ it is enough to show that for every
$g\in H_{c_{\pm}}^{0}$ we have $gU^{\pm}g^{-1}\leq\langle U^{+},U^{-}\rangle$.
As $H_{c_{\pm}}^{0}=U^{\pm}\rtimes M(H)^{0}$, where $M(H)^{0}:=\\{g\in
M(H)\;|\;g\text{ elliptic}\\}$, it is enough to verify
$gU^{\pm}g^{-1}\leq\langle U^{+},U^{-}\rangle$ only for $g\in M(H)^{0}$ and
this follows easily because $U^{\pm}$ is normal in $H_{c_{\pm}}^{0}$.

Let us now prove that $\langle U^{+},U^{-}\rangle=H^{+}$. Indeed, it is
trivial that $\langle U^{+},U^{-}\rangle\leq H^{+}$. Thus it remains to show
that for every root $\alpha$ of $\partial_{\infty}\Delta$ we have
$U_{\alpha}\leq\langle U^{+},U^{-}\rangle$. To do that we first prove that
$U^{+}$ acts transitively on the set of all chambers
$c\in\mathrm{Ch}(\partial_{\infty}\Delta)$ that are opposite $c_{+}$. In that
respect, it is enough to show that for every chamber
$c\in\mathrm{Ch}(\partial_{\infty}\Delta)$ opposite $c_{+}$ there is an
element $g\in U^{+}$ such that $g(c)=c_{-}$. By taking an apartment
$\Sigma_{1}\in\partial_{\infty}\Delta$ containing the chambers $c$ and
$c_{+}$, and using the transitivity properties of the root groups
$U_{\beta}(H)=U_{\beta}(\partial_{\infty}\Delta)$ for $\beta\in\Phi^{+}$, one
can send one by one the chambers of a gallery in $\Sigma_{1}$ from $c$ to
$c_{+}$ to chambers in $\Sigma$. This will produce an element $g\in U^{+}$
with $g(c)=c_{-}$.

Secondly, notice that by the properties of the root groups (see for example
[Ro, Chapter 6, Section 4, property (M3) and Prop. (6.14)]) we have that
$\langle U^{+},U^{-}\rangle$ acts transitively on the set of all chambers of
$\Sigma$. In particular, the spherical Weyl group associated with
$\partial_{\infty}\Delta$ is contained in $\langle U^{+},U^{-}\rangle$.

Thirdly, we prove that $\langle U^{+},U^{-}\rangle$ acts strongly transitively
on $\partial_{\infty}\Delta$. For that, it is again enough to show that for
every apartment $\Sigma_{1}$ in $\partial_{\infty}\Delta$ and every chamber
$c_{1}\in\mathrm{Ch}(\Sigma_{1})$ there is $g\in\langle U^{+},U^{-}\rangle$
such that $g(\Sigma_{1})=\Sigma$ and $g(c_{1})=c_{+}$. Then take an apartment
$\Sigma_{2}$ in $\partial_{\infty}\Delta$ with
$c_{1},c_{+}\in\mathrm{Ch}(\Sigma_{2})$. By the first step above and
considering the chamber in $\Sigma_{2}$ opposite $c_{+}$, there is an element
$g_{2}\in\langle U^{+},U^{-}\rangle$ with $g_{2}(\Sigma_{2})=\Sigma$ and
$g_{2}(c_{+})=c_{+}$, so $g_{2}(c_{1})\in\mathrm{Ch}(\Sigma)$. By applying
again the first step in combination with the second one, there is an element
$g_{1}\in\langle U^{+},U^{-}\rangle$ with $g_{1}(g_{2}(c_{1}))=g_{2}(c_{1})$
and $g_{1}(g_{2}(\Sigma_{1}))=\Sigma$. To conclude this third step we then
apply the second step in order to obtain $g_{0}\in\langle U^{+},U^{-}\rangle$
with $g_{0}(\Sigma)=\Sigma$ (not pointwise) and
$g_{0}(g_{1}(g_{2}(c_{1})))=g(g_{2}(c_{1}))=c_{+}$. Then $g:=g_{0}g_{1}g_{2}$
will do the job.

To conclude the proof of the second part of the proposition, we use the
strongly transitive action of $\langle U^{+},U^{-}\rangle$ on
$\partial_{\infty}\Delta$ and the fact that $gU_{\alpha}g^{-1}=U_{g\alpha}$
for any root $\alpha$ of $\partial_{\infty}\Delta$ and any
$g\in\mathrm{Aut}_{0}(\Delta)$ (see [AB, Lemma 7.25(1)]). ∎

###### Proof of Theorem 1.7.

By Theorem 1.5 $\mathrm{Aut}_{0}(\Delta)$ is Moufang. As moreover $\Delta$ is
irreducible, by Theorem 1.6 there exist a non-Archimedean local field $k$ and
a semisimple, (absolutely) simple algebraic group $\operatorname{\mathbb{G}}$
over $k$ and of $k$-rank $\geq 2$ such that $G=\operatorname{\mathbb{G}}(k)$
acts by type-preserving automorphisms and strongly transitively on $\Delta$.
Let $H\leq\mathrm{Aut}_{0}(\Delta)$ be a topologically simple, closed and
strongly transitive subgroup. By Theorem 1.5 the group $H$ is Moufang also.
Then by Corollary 2.3, for every root $\alpha$ of $\partial_{\infty}\Delta$ we
have $U_{\alpha}(\mathrm{Aut}_{0}(\Delta))=U_{\alpha}(H)$. By Remark 3.1 and
Proposition 3.3 we have $\overline{\langle U^{+},U^{-}\rangle}$ is normal in
$H$. As $H$ is topologically simple and $\overline{\langle
U^{+},U^{-}\rangle}$ is closed, we obtain $H=\overline{\langle
U^{+},U^{-}\rangle}=\overline{H^{+}}$. By replacing $H$ with $G$, we have
$\overline{\langle U^{+},U^{-}\rangle}$ is normal in $G$ and so also in
$G^{+}$. By [Ma, Theorem 1.5.6] one conclude that $G^{+}$ is contained in
$\overline{\langle U^{+},U^{-}\rangle}$, which is a subgroup of $G^{+}$ by
definition (recall $G^{+}$ is closed and normal in $G$). Thus
$G^{+}=\overline{\langle U^{+},U^{-}\rangle}$, so $G^{+}=H$ and we are done.

∎

##  4 Proof of Theorem 1.9

In this section we work in the setting where $\Delta$ is a locally finite
thick affine building and $G$ a closed subgroup of $\mathrm{Aut}_{0}(\Delta)$
that acts strongly transitively on $\Delta$. Fix for what follows an apartment
$\mathcal{A}$ of $\Delta$ and a special vertex $x_{0}\in\mathcal{A}$.

Let $\alpha:=\\{a_{n}\\}_{n\geq 1}$ be a sequence in $G$ that escapes every
compact subset of $G$. Then, as the space $\Delta\cup\partial_{\infty}\Delta$
is compact with respect to the cone topology, the sequence $\\{a_{n}\\}_{n\geq
1}$ admits a subsequence $\beta:=\\{a_{n_{k}}\\}_{k\geq 1}$ with the property
that $\\{a_{n_{k}}(x_{0})\\}_{k\geq 1}$ converges to an ideal point
$\xi\in\partial_{\infty}\Delta$. Notice, there is a unique minimal ideal
simplex $\sigma\subset\partial_{\infty}\Delta$ such that $\xi$ is in the
interior of $\sigma$. One can have anything in-between $\sigma$ is an ideal
chamber or $\sigma$ is an ideal vertex of the spherical building
$\partial_{\infty}\Delta$. Recall $G_{\sigma}:=\\{g\in
G\;|\;g(\sigma)=\sigma\\}$ is called a parabolic subgroup of $G$ and it is
closed. $G_{\sigma}$ is maximal when $\sigma$ is an ideal vertex in the
spherical building $\partial_{\infty}\Delta$, and minimal when $\sigma$ is an
ideal chamber of the spherical building $\partial_{\infty}\Delta$.

######  Definition 4.1.

Let $\sigma$ be an ideal simplex in $\partial_{\infty}\Delta$ and choose an
apartment $\mathcal{B}$ of $\Delta$ such that
$\sigma\subset\partial_{\infty}\mathcal{B}$. Let $x\in\mathcal{B}$ be a point.
We set
$\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B}):=\\{c\in\mathrm{Ch}(\partial_{\infty}\mathcal{B})\;|\;\sigma\subset
c\\}$, resp.,
$\operatorname{St}(x,\mathcal{B}):=\\{c\in\mathrm{Ch}(\mathcal{B})\;|\;x\text{
in the closure of }c\\}$, and call them the
$\partial_{\infty}\mathcal{B}$-star of $\sigma$, resp., the $\mathcal{B}$-star
of $x$. Notice, $\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})$,
resp., $\operatorname{St}(x,\mathcal{B})$, are both a finite set of ideal
chambers, resp., chambers. As well, the associated $\sigma$-cone in
$\mathcal{B}$ with base-point $x\in\mathcal{B}$ is defined by
$Q(\sigma,\mathcal{B},x):=\bigcup\limits_{c\in\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}Q(c,\mathcal{B},x)$,
where $Q(c,\mathcal{B},x)\subset\mathcal{B}$ is the Weyl sector emanating from
the point $x$ and having $c$ as ideal chamber. A $\sigma$-cone in
$\mathcal{B}$ is $Q(\sigma,\mathcal{B},x)$ for some $x\in\mathcal{B}$. Then we
take

$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}:=\\{g\in
G\;|\;g\text{ fixes pointwise a $\sigma$-cone in }\mathcal{B}\\}$

and it is easy to see this is a closed subgroup of $G$. Notice, the
intersection of two $\sigma$-cones in $\mathcal{B}$ contains a $\sigma$-cone
in $\mathcal{B}$.

Notice that Definition 4.1 is a weakening of Definition 2.6.

Let $\alpha:=\\{a_{n}\\}_{n\geq 1}$ be a sequence in $G$ and associated with
it consider the set

$U^{+}_{\alpha}:=\\{g\in G\;|\;\lim\limits_{n\to\infty}a_{n}^{-1}ga_{n}=e\\}$

that is a subgroup of $G$, called the positive contraction group of $\alpha$.
In the same way, but using $a_{n}ga_{n}^{-1}$ one defines the negative
contraction group $U^{-}_{\alpha}$ of $\alpha$. In general the contraction
groups are not closed in $G$.

######  Lemma 4.2.

Let $\mathcal{B}$ be an apartment of $\Delta$. Let $\alpha:=\\{a_{n}\\}_{n\geq
1}$ be a sequence in $G$ and $g$ a hyperbolic element in
$\mathrm{Stab}_{G}(\mathcal{B})$. Assume $\\{a_{n}(x_{0})\\}_{n\geq
1}\subset\mathcal{B}$ converges to an ideal point
$\xi\in\partial_{\infty}\mathcal{B}$ contained in the interior of the ideal
simplex $\sigma\subset\partial_{\infty}\mathcal{B}$.

Then $U^{+}_{\alpha}\leq
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}\leq
G_{\sigma}$, and
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ is normalized
by $g$, i.e.,
$gG_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}g^{-1}=G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$.

###### Proof.

It is clear
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}\leq
G_{\sigma}$. To prove $U^{+}_{\alpha}\leq
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ one just needs
to apply the definition of $U^{+}_{\alpha}$, to notice that each element of
$U^{+}_{\alpha}$ fixes point-wise larger and larger balls in $\Delta$ that are
centered in $\\{a_{n}(x_{0})\\}_{n\geq 1}$, and by the continuity of the
$G$-action on $\partial_{\infty}\Delta$, that element of $U^{+}_{\alpha}$
stabilizes $\sigma\subset\partial_{\infty}\mathcal{B}$. Then if an element
$h\in G$ fixes point-wise the $\mathcal{B}$-star of a special vertex $x$ in
$\mathcal{B}$ and stabilizes the ideal simplex
$\sigma\subset\partial_{\infty}\mathcal{B}$, then $h$ fixes point-wise the
combinatorial convex hall in $\mathcal{B}$ determined by
$\operatorname{St}(x,\mathcal{B})$ and $\sigma$, and this must contain the
$\sigma$-cone $Q(\sigma,\mathcal{B},x)$.

Now let us prove
$gG_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}g^{-1}=G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$.
Indeed, as $g$ is a hyperbolic element in $\mathrm{Stab}_{G}(\mathcal{B})$,
then $g(\partial_{\infty}\mathcal{B})=\partial_{\infty}\mathcal{B}$ point-
wise. Then it is clear if $h\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ fixes point-
wise the $\sigma$-cone $Q(\sigma,\mathcal{B},x)$, then
$g(Q(\sigma,\mathcal{B},x))=Q(\sigma,\mathcal{B},g(x))$ is again a
$\sigma$-cone in $\mathcal{B}$ that is point-wise fixed by $ghg^{-1}$. Thus
$gG_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}g^{-1}\subseteq
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$. In the same
way one can prove
$g^{-1}G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}g\subseteq
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$, and we are
done.

∎

The following is a generalization of [BM00b, Prop. 4.3] to the case of affine
buildings, and its proof is the same. From the point of view of finding
$H$-invariant vectors of unitary representations of $G$, with $H\leq G$ as
large as possible, it gives better results than Mautner’s phenomenon.

######  Proposition 4.3.

Let $G$ be a closed and type-preserving subgroup of $\mathrm{Aut}(\Delta)$.
Let $\sigma\subset\partial_{\infty}\Delta$ be an ideal simplex, $\mathcal{B}$
an apartment of $\Delta$ with $\sigma\subset\partial_{\infty}\mathcal{B}$, and
$x_{0}\in\mathcal{B}$ a special vertex. Let
$(\pi,\mathcal{H},\left\langle\cdot,\cdot\right\rangle)$ be a strongly
continuous unitary representation of $G_{\sigma}$ and $v,w\in\mathcal{H}$
nonzero unit vectors. Suppose there is a sequence $\\{a_{n}\\}_{n\geq
1}\subset\mathrm{Stab}_{G}(\mathcal{B})$ of hyperbolic elements such that
$\\{a_{n}(x_{0})\\}_{n\geq 1}$ converges to an ideal point $\xi$ in the
interior of $\sigma$, and $\langle\pi(a_{n})(v),w\rangle$ does not vanish at
infinity, when $n\to\infty$. Then:

  1. 1.

$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}=C_{0}\cup\bigcup\limits_{n\geq
1}a_{n}C_{0}a_{n}^{-1}$, where $C_{0}:=G_{x_{0}}\cap
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$

  2. 2.

The subspace
$\mathcal{H}^{\infty}:=\\{w\in\mathcal{H}\;|\;\mathrm{Stab}_{G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}}(w)\text{
is open in }G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}\\}$
is dense in $\mathcal{H}$

  3. 3.

There exists $w_{1}\neq 0\in\mathcal{H}$ such that
$\mathrm{Stab}_{G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}}(w_{1})$
is of finite index in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$.

###### Proof.

Let us prove 1.. First notice $C_{0}$ is a compact open subgroup of
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$. As every
$a_{n}$ is a hyperbolic element in $\mathrm{Stab}_{G}(\mathcal{B})$, then
$a_{n}(\partial_{\infty}\mathcal{B})=\partial_{\infty}\mathcal{B}$ point-wise.
Also, it is clear $C_{0}\cup\bigcup\limits_{n\geq
1}a_{n}C_{0}a_{n}^{-1}\subseteq
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$. Let $g\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ and let
$Q(\sigma,\mathcal{B},x)$ be a $\sigma$-cone in $\mathcal{B}$ point-wise fixed
by $g$. As $Q(\sigma,\mathcal{B},x)$ is a convex cone and because
$\\{a_{n}(x_{0})\\}_{n\geq 1}$ converges to $\xi\in\sigma$, there exists $N>0$
such that $a_{n}(x_{0})\in Q(\sigma,\mathcal{B},x)$ for every $n\geq N$. Then
$Q(\sigma,\mathcal{B},a_{n}(x_{0}))\subset Q(\sigma,\mathcal{B},x)$ for $n\geq
N$, and so $g\in a_{n}C_{0}a_{n}^{-1}$.

Let us prove 2.. Fix a decreasing sequence $\\{K_{n}\\}_{n\geq 1}$ of compact
open subgroups of
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ such that
$\bigcap\limits_{n\geq 1}K_{n}=\\{e\\}$. Such sequence exists. Then the
sequence $\\{f_{n}\\}_{n\geq 1}$, where
$f_{n}:=\frac{1}{m(K_{n})}\cdot\chi_{K_{n}}$ is an $L^{1}$-approximation of
the identity $e$. Hence $\lim\limits_{n\to\infty}\pi(f_{n})(w_{1})=w_{1}$ for
every $w_{1}\in\mathcal{H}$. As
$\pi(f_{n})(w_{1})=\int\limits_{K_{n}}\pi(k)(w_{1})dk\in\mathcal{H}^{\infty}$,
for every $w_{1}\in\mathcal{H}$, we obtain the conclusion.

Let us prove 3.. As by hypothesis $\langle\pi(a_{n})(v),w\rangle$ does not
vanish at infinity, when $n\to\infty$, by 2. there exist nonzero
$u_{1},u_{2}\in\mathcal{H}^{\infty}$ with
$\langle\pi(a_{n})(u_{1}),u_{2}\rangle$ not vanishing at infinity either. Then
there is a subsequence $\\{n_{i}\\}_{i\geq 1}$ with
$n_{i}\xrightarrow[i\to\infty]{}\infty$ and $w_{1}\in\mathcal{H}$, $w_{1}\neq
0$ such that $\\{\pi(a_{n_{i}})(u_{1})\\}_{i\geq 1}$ converges weakly to
$w_{1}$.

Let $K\leq C_{0}$ be an open subgroup (thus of finite index and closed in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$) for which
$\pi(k)(u_{1})=u_{1}$, for every $k\in K$. Then $\pi(a_{n_{i}})(u_{1})$ is
fixed by $a_{n_{i}}Ka_{n_{i}}^{-1}$. By passing to a subsequence, one may
assume the sequence of closed subgroups $\\{a_{n_{i}}Ka_{n_{i}}^{-1}\\}_{i\geq
1}$ converges in the Chabauty topology to a closed subgroup $L\leq
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$. We claim
$w_{1}$ is $L$-invariant and $L$ is of finite index in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$.

Claim 1: $w_{1}$ is $L$-invariant. Indeed, let $\ell\in L$. Then there is
$m_{i}\in a_{n_{i}}Ka_{n_{i}}^{-1}$, for every $i\geq 1$, such that
$\lim\limits_{i\to\infty}m_{i}=\ell$. We have

$\begin{split}\langle\pi(\ell)(w_{1}),w_{1}\rangle-\langle
w_{1},w_{1}\rangle&=(\langle\pi(\ell)(w_{1}),w_{1}\rangle-\langle\pi(\ell)(w_{1}),\pi(a_{n_{i}})(u_{1})\rangle)\\\
&\;\;+(\langle\pi(\ell)(w_{1}),\pi(a_{n_{i}})(u_{1})\rangle-\langle\pi(m_{i})(w_{1}),\pi(a_{n_{i}})(u_{1})\rangle)\\\
&\;\;+(\langle w_{1},\pi(a_{n_{i}})(u_{1})\rangle-\langle
w_{1},w_{1}\rangle).\\\ \end{split}$ (1)

Notice, $m_{i}\in a_{n_{i}}Ka_{n_{i}}^{-1}$ implies
$\langle\pi(m_{i})(w_{1}),\pi(a_{n_{i}})(u_{1})\rangle=\langle
w_{1},\pi(a_{n_{i}})(u_{1})\rangle)$. Also the first and the last summand from
equality (1) converger to zero since $\\{\pi(a_{n_{i}})(u_{1})\\}_{i\geq 1}$
converges weakly to $w_{1}$. The second summand also converges to zero since
it is bounded by $||\pi(\ell)(w_{1})-\pi(m_{i})(w_{1})||\cdot||u_{1}||$ and
$\lim\limits_{i\to\infty}m_{i}=\ell$. The Claim 1 follows.

Claim 2: the subgroup $L$ is of finite index in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$. Indeed, let
$d\in\mathbb{N}$ be the index of $K$ in $C_{0}$ and pick
$x_{1},\cdots,x_{r}\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ with $r>d$.
Since
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}=\bigcup\limits_{i\geq
1}a_{n_{i}}C_{0}a_{n_{i}}^{-1}$, there exists $i_{0}$ such that
$\\{x_{1},\cdots x_{r}\\}\subset a_{n_{i}}C_{0}a_{n_{i}}^{-1}$ for every
$i\geq i_{0}$. Since $r>d$, for every $i\geq i_{0}$, there exist $j_{i}\neq
k_{i}$ such that $x_{j_{i}}^{-1}x_{k_{i}}\in a_{n_{i}}Ka_{n_{i}}^{-1}$. As
$r<+\infty$, there exist $j\neq k$ such that $x_{j}^{-1}x_{k}\in
a_{n_{i}}Ka_{n_{i}}^{-1}$ for infinitely many $i\geq i_{0}$. This implies
$x_{j}^{-1}x_{k}\in L$ and shows that the index of $L$ in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{B})}^{0}$ is at most
$d$. This proves the claim and the proposition. ∎

######  Lemma 4.4.

(See [Cio, Lemma 4.12]) Let $G$ be a closed, strongly transitive and type-
preserving subgroup of $\mathrm{Aut}(\Delta)$, and
$\sigma\subset\partial_{\infty}\Delta$ an ideal simplex. Let
$\mathcal{A},\mathcal{B}$ be two apartments in $\Delta$ such that
$\mathcal{A}\cap\mathcal{B}$ is a half-apartment containing
$\mathrm{St}(\sigma,\partial_{\infty}\mathcal{A})$ in its boundary at
infinity. Then:

  1. 1.

There is $g\in G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$
mapping $\mathcal{B}$ to $\mathcal{A}$ and fixing point-wise
$\mathcal{A}\cap\mathcal{B}$

  2. 2.

For every wall $M$ of $\mathcal{A}$ there is $g\in G$ that stabilizes
$\mathcal{A}$ and acts on $\mathcal{A}$ as the reflection through the wall
$M$.

  3. 3.

For every wall $M$ of $\mathcal{A}$ there is a hyperbolic element
$g\in\mathrm{Stab}_{G}(\mathcal{A})$ whose translation axis is perpendicular
to $M$.

###### Proof.

Part 1. of the lemma follows by the strong transitivity of $G$ on $\Delta$,
and by choosing a chamber $c$ in the interior of the half-apartment
$\mathcal{A}\cap\mathcal{B}$. Then there is $g\in G$ such that
$g(\mathcal{B})=\mathcal{A}$ and $g(c)=c$. In particular, $g$ fixes point-wise
the half-apartment $\mathcal{A}\cap\mathcal{B}$, and as well its ideal
boundary containing $\mathrm{St}(\sigma,\partial_{\infty}\mathcal{A})$. This
concludes $g\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$.

To prove part 2., let $M$ be a wall of $\mathcal{A}$. Let $H$ and $H^{\prime}$
be the two half-apartments of $\mathcal{A}$ determined by $M$. Since $\Delta$
is thick, there is a half-apartment $H^{\prime\prime}$ such that $H\cup
H^{\prime\prime}$ and $H^{\prime}\cup H^{\prime\prime}$ are both apartments of
$\Delta$. By the first part of the lemma there are $u,v,w\in G$ such that $u$
fixes $H$ point-wise and maps $H^{\prime}$ to $H^{\prime\prime}$, the elements
$v,w$ both fix $H^{\prime}$ point-wise and
$v(H^{\prime\prime})=H,w(H)=u^{-1}(H^{\prime})$. Set $g=vuw$ and by
construction $g$ fixes point-wise the wall $M$ and
$g(H)=H^{\prime},g(H^{\prime})=H$.

To obtain the last part of the lemma one just needs to apply 2. for example,
for two consecutive parallel walls of $\mathcal{A}$. The conclusion follows. ∎

We also record the following lemma. Still, this is not used further.

######  Lemma 4.5.

Let $G$ be a closed, strongly transitive and type-preserving subgroup of
$\mathrm{Aut}(\Delta)$. Let $\mathcal{A}$ be an apartment of $\Delta$,
$\sigma$ a simplex in $\partial_{\infty}\mathcal{A}$, and $M$ a wall in
$\mathcal{A}$. Suppose
$\mathrm{St}(\sigma,\partial_{\infty}\mathcal{A})\subset\partial_{\infty}H$,
where $H$ is one of the half-apartments of $\mathcal{A}$ associated with $M$.
Let $r\in\mathrm{Stab}_{G}(\mathcal{A})$ be a reflection on $\mathcal{A}$
through the wall $M$.

Then for every hyperbolic element $g\in\mathrm{Stab}_{G}(\mathcal{A})$ of even
translation length, and having its translation axis perpendicular to $M$ and
its repelling end-point in $\partial_{\infty}H$, the double coset
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}rgG_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$
contains elliptic elements that fix point-wise a half-apartment of
$\mathcal{A}$ associated with an $M$-parallel wall between $M$ and
$g^{-1}(M)$, and not containing $\sigma$ in its ideal boundary.

###### Proof.

Let $g$ be a hyperbolic element in $\mathrm{Stab}_{G}(\mathcal{A})$ of even
translation length, and having its translation axis perpendicular to $M$ and
its repelling end-point in $\partial_{\infty}H$.

As $g$ has even translation length, let $M^{\prime}$ be the $M$-parallel wall
in $\mathcal{A}$ at the half distance between $M$ and $g^{-1}(M)$. Notice the
wall $M^{\prime}$ is in the half-apartment $H$. Denote by $H^{\prime}$ the
half-apartment of $\mathcal{A}-M^{\prime}$ having
$\mathrm{St}(\sigma,\partial_{\infty}\mathcal{A})$ in its ideal boundary. By
Lemma 4.4 there is $h\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$ fixing
$H^{\prime}$ point-wise and moving away the half apartment
$\mathcal{A}-H^{\prime}$. Then apply $g$ and so the half-apartment
$gh(\mathcal{A}-H^{\prime})$ intersects $\mathcal{A}$ only in the wall
$g(M^{\prime})$. Apply the reflection $r$, and by the above construction
$rg(M^{\prime})=M^{\prime}$. Apply again an element $h^{\prime}\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$ that fixes
$H^{\prime}$ point-wise and moves the half-apartment
$rgh(\mathcal{A}-H^{\prime})$ back to $\mathcal{A}-H^{\prime}$. We have
$h^{\prime}rgh(\mathcal{A}-H^{\prime})=\mathcal{A}-H^{\prime}$ point-wise.
Notice $h^{\prime}rgh(H^{\prime})\neq H^{\prime}$, and we are done. ∎

######  Lemma 4.6.

Let $G$ be a closed, strongly transitive and type-preserving subgroup of
$\mathrm{Aut}(\Delta)$. Let $\mathcal{A}$ be an apartment of $\Delta$ and
$\sigma$ a simplex in $\partial_{\infty}\mathcal{A}$. Then there exist a
sequence of half-apartments $\\{H_{n}\\}_{n\geq 1}$ of $\mathcal{A}$ and
elements $\\{k_{n}\\}_{n\geq 1}\subset G_{x_{0}}$ such that:

  1. 1.

$H_{n}\subsetneq H_{n+1}$, and $x_{0}\in H_{n}$, for every $n\geq 1$

  2. 2.

$\mathrm{St}(\sigma,\partial_{\infty}\mathcal{A})\subset\partial_{\infty}(\mathcal{A}-H_{1})$

  3. 3.

$\lim\limits_{n\to\infty}k_{n}=e$, and for every $n\geq 1$ one has
$k_{n}(H_{n})=H_{n}$ and $k_{n}(\mathcal{A}-H_{n})\cap\mathcal{A}$ is a wall.

###### Proof.

One uses Lemma 4.4 and the fact that, when $\Delta$ is irreducible, by
Theorems 1.5 and 1.7, and Proposition 3.3, $G$ has many and enough elements. ∎

######  Lemma 4.7.

Let $G$ be a closed, strongly transitive and type-preserving subgroup of
$\mathrm{Aut}(\Delta)$. Let $\mathcal{A}$ be an apartment of $\Delta$,
$\sigma$ a simplex in $\partial_{\infty}\mathcal{A}$ and $M$ a wall in
$\mathcal{A}$. Suppose
$\mathrm{St}(\sigma,\partial_{\infty}\mathcal{A})\subset\partial_{\infty}H$,
where $H$ is one of the two half-apartments of $\mathcal{A}$ associated with
$M$. Let $k\in G$ such that $k(\mathcal{A}-H)=\mathcal{A}-H$ point-wise and
$k(H)\cap\mathcal{A}=M$. Then for every $\ell\in\mathbb{N}^{*}$ there exists
$g\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}kG_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$
that is hyperbolic, of translation axis perpendicular to $M$, containing
$\mathcal{A}-H$ in its translation apartment, of attracting end-point in
$\partial_{\infty}(\mathcal{A}-H)$, and of translation length $2\ell$.

###### Proof.

Let $\ell\in\mathbb{N}^{*}$. Choose the $M$-parallel wall $M^{\prime}$
contained in $H$ such that
$\operatorname{dist}_{\mathcal{A}}(M,M^{\prime})=\ell$. Let $H^{\prime}$ be
the half-apartment of $\mathcal{A}$ with wall-boundary $M^{\prime}$ and
containing $\mathrm{St}(\sigma,\partial_{\infty}\mathcal{A})$ in its ideal
boundary $\partial_{\infty}H^{\prime}$. By Lemma 4.4 there is $h\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$ fixing
$H^{\prime}$ point-wise and moving away the half apartment
$\mathcal{A}-H^{\prime}$. Then $\mathcal{B}:=(\mathcal{A}-H^{\prime})\cup
h(\mathcal{A}-H^{\prime})$ is again an apartment of $\Delta$ that contains
$\mathcal{A}-H$ as a half-apartment. Let
$H^{\prime\prime}:=\mathcal{B}-(\mathcal{A}-H)$ and notice this half-apartment
has $M$ as a boundary wall. Apply $k$ to $H^{\prime\prime}$, and so by our
construction $k(H^{\prime\prime})\cap\mathcal{A}=M$. Then $\mathcal{C}:=H\cup
k(H^{\prime\prime})$ is again an apartment in $\Delta$. We apply again Lemma
4.4 and find an element $h^{\prime}\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$ with
$h^{\prime}(H)=H$ point-wise and
$h^{\prime}k(H^{\prime\prime})\subset\mathcal{A}$. By tracking the orientation
during the construction, one notices $h^{\prime}kh$ is a hyperbolic element of
translation axis perpendicular to the wall $M$, containing $\mathcal{A}-H$ in
its translation apartment, and with attracting end-point in
$\partial_{\infty}(\mathcal{A}-H)$. The translation length of $h^{\prime}kh$
is exactly $2\ell$. ∎

Let $G$ be a closed, strongly transitive and type-preserving subgroup of
$\mathrm{Aut}(\Delta)$. Let $\mathcal{A}$ be an apartment of $\Delta$, an
ideal chamber $c$ of the spherical apartment $\partial_{\infty}\mathcal{A}$,
and $x_{0}$ a special vertex of $\mathcal{A}$. Then we have the well known
polar decomposition $G=G_{x_{0}}AG_{x_{0}}$, where
$A:=\\{g\in\mathrm{Stab}_{G_{0}}(\mathcal{A})\;|\;g\text{
hyperbolic}\\}\cup\\{e\\}$ is the subgroup of euclidean translations in the
apartment $\mathcal{A}$ (for a proof see for example [Cio, Lemma 4.7]). As
well, by the well known Bruhat decomposition we have
$G=G_{c}WG_{c}=G_{c}^{0}\mathrm{Stab}_{G}(\mathcal{A})G_{c}^{0}$, where $W$ is
the finite Weyl group associated with $\Delta$ and its elements can be lifted
(not uniquely) to elements in $\mathrm{Stab}_{G}(\mathcal{A})$.

In what follows we will consider unitary representations $(\pi,\mathcal{H})$
of $G$ that are strongly continuous. The idea of the proof of Theorem 1.9 is
the same as in [BM00b].

###### Proof of Theorem 1.9.

Fix a unitary representation $(\pi,\mathcal{H})$ of $G$ without non-zero
$G$-invariant vectors. Suppose there exist two non-zero vectors
$v_{1},v_{2}\in\mathcal{H}$ such that the $(v_{1},v_{2})$-matrix coefficient
does not vanish at $\infty$. As $G=G_{x_{0}}AG_{x_{0}}$ there exist a sequence
$\\{a_{n}\\}_{n\geq 1}\subset A$ and non-zero vectors
$v_{0},w_{0}\in\mathcal{H}$ such that $a_{n}\xrightarrow[n\to\infty]{}\infty$
and $|\langle\pi(a_{n})(v_{0}),w_{0}\rangle|$ does not converges to $0$, when
$n\to\infty$ (see for example [Cio, Lemma 2.9]).

As $a_{n}\xrightarrow[n\to\infty]{}\infty$ and
$\Delta\cup\partial_{\infty}\Delta$ is a compact space with respect to the
cone topology, by extracting a subsequence, we can suppose in addition
$\\{a_{n}(x_{0})\\}_{n\geq 1}$ converges to an ideal point
$\xi\in\partial_{\infty}\mathcal{A}$ contained in the interior of a minimal
ideal simplex $\sigma\subset\partial_{\infty}\mathcal{A}$. Then apply
Proposition 4.3 to our sequence $\\{a_{n}\\}_{n\geq
1}\subset\mathrm{Stab}_{G}(\mathcal{A})$ of hyperbolic elements and the non-
zero vectors $v_{0},w_{0}\in\mathcal{H}$ to obtain a non-zero vector
$w\in\mathcal{H}$ such that
$L:=\mathrm{Stab}_{G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}}(w)$
is of finite index in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$.

Then consider the $(w,w)$-matrix coefficient on $G$, $f:G\to\mathbb{C}$, given
by $f(g):=\langle\pi(g)(w),w\rangle$. This map $f$ is continuous with respect
to the topology on $G$ and $L$-bi-invariant (i.e., $f(\ell_{1}g\ell_{2})=f(g)$
for every $\ell_{1},\ell_{2}\in L$).

Choose some ideal chamber $c$ with $\sigma\subset
c\in\mathrm{Ch}(\partial_{\infty}\mathcal{A})$. Then we have
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}\leq G_{c}$. If
$\sigma$ is just a sub-simplex of $c$, then
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$ is of
infinite index in $G_{c}^{0}$, the latter being a normal subgroup of $G_{c}$.
Still, one can use Bruhat decomposition
$G=G_{c}WG_{c}=G_{c}^{0}\mathrm{Stab}_{G}(\mathcal{A})G_{c}^{0}$ and write
$G=LHL$, where $H$ is a subset of $G$ containing many elements from
$\mathrm{Stab}_{G}(\mathcal{A})$ (e.g., hyperbolic elements), as well as many
other elements.

By Lemma 4.6, consider a sequence $\\{k_{n}\\}_{n\geq 1}\subset G_{x_{0}}$ and
a sequence of half-apartments $\\{H_{n}\\}_{n\geq 1}$ in $\mathcal{A}$ with
the required properties. By the continuity of the map $f$ we have

$f(Lk_{n}L)\xrightarrow[n\to\infty]{}f(L)=1.$

Let $j\in\mathbb{N}$. By Lemma 4.7, applied to every $k_{n}$, the double coset
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}k_{n}G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$
contains a hyperbolic element $g_{n}=h_{n}k_{n}h_{n}^{\prime}$, with
$h_{n},h_{n}^{\prime}\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$, of
translation length $2j$, containing the half-apartment $H_{n}$ in its
translation apartment, of translation axis perpendicular to boundary wall of
$H_{n}$, and of attracting end-point in
$\partial_{\infty}(H_{n})\subset\partial_{\infty}\mathcal{A}$. Then
$\\{g_{n}=h_{n}k_{n}h_{n}^{\prime}\\}_{n\geq 1}$ admits a subsequence, by
simplicity denoted by $\\{g_{n}\\}_{n\geq 1}$, that converges to an element
$g_{j}\in A$ that is hyperbolic, of attracting endpoint in
$\partial_{\infty}(H_{1})\subset\partial_{\infty}\mathcal{A}$, of translation
length $2j$, and whose translation axis is perpendicular to the boundary wall
of $H_{1}$.

Moreover, as $L$ has finite index in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$ and
$h_{n},h_{n}^{\prime}\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$, by passing to
a subsequence $\\{g_{n_{i}}\\}_{i\geq 1}$ we can write
$h_{n_{i}}=\sigma\ell_{n_{i}}$,
$h_{n_{i}}^{\prime}=\ell_{n_{i}}^{\prime}\sigma^{\prime}$, with
$\ell_{n_{i}},\ell_{n_{i}}^{\prime}\in L$ and $\sigma,\sigma^{\prime}\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$. Then

$\lim\limits_{i\to\infty}\ell_{n_{i}}k_{n_{i}}\ell_{n_{i}}^{\prime}=\sigma^{-1}g_{j}(\sigma^{\prime})^{-1}$

and because
$g_{j}G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}g_{j}^{-1}=G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$
(see Lemma 4.2) we can write further as

$\lim\limits_{i\to\infty}\ell_{n_{i}}k_{n_{i}}\ell_{n_{i}}^{\prime}=g_{j}\sigma_{j},\text{
with }\sigma_{j}\in
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}.$

Then as well by the continuity of the map $f$ we have

$f(Lk_{n}L)\xrightarrow[n\to\infty]{}f(Lg_{j}\sigma_{j}L)$

and therefore $f(Lg_{j}\sigma_{j}L)=1=\langle\pi(g_{j}\sigma_{j})(w),w\rangle$
for every $j\in\mathbb{N}$. We have the equality
$\langle\pi(g_{j}\sigma_{j})(w),w\rangle=\langle w,w\rangle$ if and only if
$w$ is $g_{j}\sigma_{j}$-invariant, for every $j\in\mathbb{N}$. Since $L$ has
finite index in
$G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}$, we can
choose $\\{\sigma_{j}\\}_{j\in\mathbb{N}}\subset
G_{\operatorname{St}(\sigma,\partial_{\infty}\mathcal{A})}^{0}/L$ that is a
finite set. As $g_{j}g_{j+i}^{-1}$ is still a hyperbolic element of $A$, for
every $i,j\in\mathbb{N}$, one can conclude there is a hyperbolic element $a\in
A-\\{e\\}$ such that $w$ is $a$-invariant. Then by standard methods as
Mautner’s phenomenon (see for example [Cio, Prop. 2.12]) applied to the
sequence $\\{a^{n}\\}_{n\geq 1}$, resp., $\\{a^{-n}\\}_{n\geq 1}$, one has $w$
is $U_{a}^{-}$ and $U_{a}^{+}$-invariant. If in addition $G$ is topologically
simple, by [Cio, Corollary 4.18], the non-zero vector $w$ is $G$-invariant.
This is a contradiction with our assumption at the beginning of the proof, and
so the theorem is proven. ∎

## References