arXiv:1711.01057v2 [] 13 Nov 2017

Open subgroups of the automorphism group of a
right-angled building
Tom De Medts

Ana C. Silva

November 15, 2017

Abstract
We study the group of type-preserving automorphisms of a rightangled building, in particular when the building is locally ﬁnite. Our
aim is to characterize the proper open subgroups as the ﬁnite index
closed subgroups of the stabilizers of proper residues.
One of the main tools is the new notion of ﬁrm elements in a rightangled Coxeter group, which are those elements for which the ﬁnal
letter in each reduced representation is the same. We also introduce
the related notions of ﬁrmness for arbitrary elements of such a Coxeter
group and n-ﬂexibility of chambers in a right-angled building. These
notions and their properties are used to determine the set of chambers
ﬁxed by the ﬁxator of a ball. Our main result is obtained by combining these facts with ideas by Pierre-Emmanuel Caprace and Timothée
Marquis in the context of Kac–Moody groups over ﬁnite ﬁelds, where
we had to replace the notion of root groups by a new notion of root
wing groups.

1

Introduction

A Coxeter group is right-angled if the entries of its Coxeter matrix are all
equal to 1, 2 or ∞ (see Deﬁnition 2.1 below for more details). A right-angled
building is a building for which the underlying Coxeter group is right-angled.
The most prominent examples of right-angled buildings are trees. To some
extent, the combinatorics of right-angled Coxeter groups and right-angled
buildings behave like the combinatorics of trees, but in a more complicated
and therefore in many aspects more interesting fashion.
Right-angled buildings have received attention from very diﬀerent perspectives. One of the earlier motivations for their study was the connection
with lattices; see, for instance, [RR06, Tho06, TW11, KT12, CT13]. On the
other hand, the automorphism groups of locally ﬁnite right-angled buildings are totally disconnected locally compact (t.d.l.c.) groups, and their
full automorphism group was shown to be an abstractly simple group by
Pierre-Emmanuel Caprace in [Cap14], making these groups valuable in the
1

study of t.d.l.c. groups. Caprace’s work also highlighted important combinatorial aspects of right-angled buildings; in particular, his study of parallel residues and his notion of wings (see Deﬁnition 3.6 below) are fundamental tools. From this point of view, we have, in a joint work with
Koen Struyve, introduced and investigated universal groups for right-angled
buildings; see [DMSS16]. More recently, Andreas Baudisch, Amador MartinPizarro and Martin Ziegler have studied right-angled buildings from a modeltheoretic point of view; see [BMPZ17].
In this paper, we continue the study of right-angled buildings in a combinatorial and topological fashion. In particular, we introduce some new
tools in right-angled Coxeter groups and we study the (full) automorphism
group of right-angled buildings. Our main goal is to characterize the proper
open subgroups of the automorphism group of a locally ﬁnite semi-regular
right-angled building as the closed ﬁnite index subgroups of the stabilizer of
a proper residue; see Theorem 4.29 below.
The ﬁrst tool we introduce is the notion of firm elements in a rightangled Coxeter group: these are the elements with the property that every
possible reduced representation of that element ends with the same letter
(see Deﬁnition 2.10 below), i.e., the last letter cannot be moved away by
elementary operations. If an element of the Coxeter group is not ﬁrm, then
we deﬁne its firmness as the maximal length of a ﬁrm preﬁx.
This notion will be used to deﬁne the concepts of firm chambers in a
right-angled building and of n-flexibility of chambers with respect to another
chamber; this then leads to the notion of the n-flex of a given chamber. See
Deﬁnition 3.9 below.
A second new tool is the concept of a root wing group, which we deﬁne in
Deﬁnition 4.6. Strictly speaking, this is not a new deﬁnition since the root
wing groups are deﬁned as wing ﬁxators, and as such they already appear
in the work of Caprace [Cap14]. However, we associate such a group to
a root in an apartment of the building, and we explore the fact that they
behave very much like root subgroups in groups of a more algebraic nature,
such as automorphism groups of Moufang spherical buildings or Kac–Moody
groups.
Outline of the paper. In Section 2, we provide the necessary tools for
right-angled Coxeter groups. In Section 2.1, we recall the notion of a poset
≺w that we can associate to any word w in the generators, introduced in
[DMSS16]. Section 2.2 introduces the concepts of ﬁrm elements and the
ﬁrmness of elements in a right-angled Coxeter group. Our main result in
that section is the fact that long elements cannot have a very low ﬁrmness;
see Theorem 2.18.
Section 3 collects combinatorial facts about right-angled buildings. After recalling the important notions of parallel residues and wings, due to
2

Caprace [Cap14], in Section 3.1, we proceed in Section 3.2 to introduce the
notion of chambers that are n-ﬂexible with respect to another chamber and
the notion of the square closure of a set of chambers (which is based on
results from [DMSS16]); see Deﬁnitions 3.9 and 3.12. Our main result in
Section 3 is Theorem 3.13, showing that the square closure of a ball of radius
n around a chamber c0 is precisely the set of chambers that are n-ﬂexible
with respect to c0 .
In Section 4, we study the automorphism group of a semi-regular rightangled building. We begin with a short Section 4.1 that uses the results of
the previous sections to show that the set of chambers ﬁxed by a ball ﬁxator
is bounded; see Theorem 4.4. In Section 4.2, we associate a root wing group
Uα to each root (Deﬁnition 4.6), we show that Uα acts transitively on the
set of apartments through α (Proposition 4.7) and we adopt some facts
from [CM13] to the setting of root wing groups.
We then continue towards our characterization of the open subgroups
of the full automorphism group of a semi-regular locally ﬁnite right-angled
building. Our ﬁnal result is Theorem 4.29 showing that every proper open
subgroup is a ﬁnite index subgroup of the stabilizer of a proper residue.
We distinguish between the case where the open subgroup is compact (Section 4.3) and non-compact (Section 4.4). In the compact case, we provide
a characterization that remains valid for right-angled buildings that are not
locally ﬁnite, and we use our knowledge about the ﬁxed-point set of ball
ﬁxators; see Proposition 4.15. In the non-compact case, we have to restrict
to locally ﬁnite buildings. We follow, to a very large extent, the strategy
taken by Pierre-Emmanuel Caprace and Timothée Marquis in [CM13] in
their study of open subgroups of Kac–Moody groups over ﬁnite ﬁelds; in
particular, we show that an open subgroup of Aut(∆) contains suﬃciently
many root wing groups, and much of the subtleties of the proof go into
determining precisely the types of the root groups contained in the open
subgroup, which will then, in turn, pin down the residue, the stabilizer of
which contains the given open subgroup as a ﬁnite index subgroup.
In the ﬁnal Section 5, we mention two applications of our main theorem. The ﬁrst is a rather immediate corollary, namely the fact that the
automorphism group of a semi-regular locally ﬁnite right-angled building is
a Noetherian group; see Proposition 5.3. The second application shows that
every open subgroup of the automorphism group is the reduced envelope of
a cyclic subgroup; see Proposition 5.6.
Acknowledgments. This paper would never have existed without the
help of Pierre-Emmanuel Caprace. Not only did he suggest the study of
open subgroups of the automorphism group of right-angled buildings to us;
we also beneﬁted a lot from discussions with him.
We also thank the Research Foundation in Flanders (F.W.O.-Vlaanderen)

3

for their support through the project “Automorphism groups of locally ﬁnite
trees” (G011012N).

2

Right-angled Coxeter groups

We begin by recalling some basic deﬁnitions and facts about Coxeter groups.
Definition 2.1. (i) A Coxeter group is a group W with generating set
S = {s1 , . . . , sn } and with presentation
W = hs ∈ S | (st)mst = 1i
where mss = 1 for all s ∈ S and mst = mts ≥ 2 for all i 6= j.
It is allowed that mst = ∞, in which case the relation involving st
is omitted. The pair (W, S) is called a Coxeter system of rank n.
The matrix M = (msi sj ) is called the Coxeter matrix of (W, S). The
Coxeter matrix is often conveniently encoded by its Coxeter diagram,
which is a labeled graph with vertex set S where two vertices are joined
by an edge labeled mst if and only if mst ≥ 3.
(ii) A Coxeter system (W, S) is called right-angled if all entries of the Coxeter matrix are 1, 2 or ∞. In this case, we call the Coxeter diagram Σ
of W a right-angled Coxeter diagram; all its edges have label ∞.
Definition 2.2. Let (W, S) be a Coxeter system and let J ⊆ S.
(i) We deﬁne WJ := hs | s ∈ Ji ≤ W and we call this a standard
parabolic subgroup of W . It is itself a Coxeter group, with Coxeter
system (WJ , J). Any conjugate of a standard parabolic subgroup WJ
is called a parabolic subgroup of W .
(ii) The subset J ⊆ S is called a spherical subset if WJ is ﬁnite. When
(W, S) is right-angled, J is spherical if and only if |st| ≤ 2 for all
s, t ∈ J.
(iii) The subset J ⊆ S is called essential if each irreducible component
of J is non-spherical. In general, the union J0 of all irreducible nonspherical components of J is called the essential component of J.
If P is a parabolic subgroup of W conjugate to some WJ , then the
essential component P0 of P is the corresponding conjugate of WJ0 ,
where J0 is the essential component of J. Observe that P0 has ﬁnite
index in P .
(iv) Let E ⊆ W . We deﬁne the parabolic closure of E, denoted by Pc(E),
as the smallest parabolic subgroup of W containing E.
Lemma 2.3 ([CM13, Lemma 2.4]). Let H1 ≤ H2 be subgroups of W . If H1
has finite index in H2 , then Pc(H1 ) has finite index in Pc(H2 ).
4

2.1

A poset of reduced words

Let Σ = (W, S) be a right-angled Coxeter system and let MS be the free
monoid over S, the elements of which we refer to as words. Notice that there
is an obvious map MS → W denoted by w 7→ w; if w ∈ MS is a word, then
its image w under this map is called the element represented by w, and the
word w is called a representation of w. For w1 , w2 ∈ MS , we write w1 ∼ w2
when w1 = w2 . By some slight abuse of notation, we also say that w2 is a
representation of w1 (rather than a representation of w1 ).
Definition 2.4. A Σ-elementary operation on a word w ∈ MS is an operation of one of the following two types:
(1) Delete a subword of the form ss, with s ∈ S.
(2) Replace a subword st by ts if mst = 2.
A word w ∈ MS is called reduced (with respect to Σ) if it cannot be shortened
by a sequence of Σ-elementary operations.
Clearly, applying elementary operations on a word w does not alter its
value in W . Conversely, if w1 ∼ w2 for two words w1 , w2 ∈ MS , then w1
can be transformed into w2 by a sequence of Σ-elementary operations. The
number of letters in a reduced representation of w ∈ W is called the length
of w and is denoted by l(w). Tits proved in [Tit69] (for arbitrary Coxeter
systems) that two reduced words represent the same element of W if and
only if one can be obtained from the other by a sequence of elementary
operations of type (2) (or rather its generalization to all values for mst ).
Definition 2.5. Let w = s1 s2 · · · sℓ ∈ MS . If σ ∈ Sym(ℓ), then we let
σ.w be the word obtained by permuting the letters in w according to the
permutation σ, i.e.,
σ.w := sσ(1) sσ(2) · · · sσ(ℓ) .
In particular, if w′ is obtained from w by applying an elementary operation of
type (2) replacing si si+1 by si+1 si , then σ.w = w′ for σ = (i i+ 1) ∈ Sym(ℓ).
In this case, si and si+1 commute and we call σ = (i i + 1) a w-elementary
transposition.
In this way, we can associate an elementary transposition to each Σ-elementary operation of type (2). It follows that two reduced words w and w′
represent the same element of W if and only if
w′ = (σn · · · σ1 ).w, where each σi is a
(σi−1 · · · σ1 ).w-elementary transposition,
i.e., if w′ is obtained from w by a sequence of elementary transpositions.

5

Definition 2.6. If w ∈ MS is a reduced word of length ℓ, then we deﬁne
Rep(w) := {σ ∈ Sym(ℓ) | σ = σn · · · σ1 , where each σi is a
(σi−1 · · · σ1 ).w-elementary transposition}.
In other words, the set Rep(w) consists of the permutations of ℓ letters which
give rise to reduced representations of w.
We now deﬁne a partial order ≺w on the letters of a reduced word w in
MS with respect to Σ.
Definition 2.7 ([DMSS16, Deﬁnition 2.6]). Let w = s1 · · · sℓ be a reduced
word of length ℓ in MS and let Iw = {1, . . . , ℓ}. We deﬁne a partial order
“≺w ” on Iw as follows:
i ≺w j ⇐⇒ σ(i) > σ(j) for all σ ∈ Rep(w).
Note that i ≺w j implies that i > j. As a mnemonic, one can regard
j ≻w i as “j → i”, i.e., the generator si comes always after the generator sj
regardless of the reduced representation of w.
We point out a couple of basic but enlightening consequences of the
deﬁnition of this partial order.
Observation 2.8. Let w = s1 · · · si · · · sj · · · sℓ be a reduced word in MS
with respect to a right-angled Coxeter diagram Σ.
(i) If |si sj | = ∞, then i ≻w j.
The converse is not true. Indeed, suppose there is i < k < j such that
|si sk | = ∞ and |sk sj | = ∞. Then i ≻w j, independently of whether
|si sj | = 2 or ∞.
(ii) If i 6≻w j, then by (i), it follows that |si sj | = 2 and, moreover, for each
k ∈ {i + 1, . . . , j − 1}, either |si sk | = 2 or |sk sj | = 2 (or both).
(iii) On the other hand, if sj and sj+1 are consecutive letters in w, then
|sj sj+1 | = ∞ if and only if j ≻w j + 1.
Lemma 2.9 ([DMSS16, Lemma 2.8]). Let w = w1 · si · · · sj · w2 ∈ MS be a
reduced word. If i 6≻w j, then there exist two reduced representations of w
of the form
w1 · · · si sj · · · w2 and w1 · · · sj si · · · w2 ,
i.e., the positions of si and sj can be exchanged using only elementary operations on the generators {si , si+1 , . . . , sj−1 , sj }, without changing the prefix
w1 and the suffix w2 .

6

2.2

Firm elements of right-angled Coxeter groups

In this section we deﬁne ﬁrm elements in a right-angled Coxeter group W
and we introduce the concept of ﬁrmness to measure “how ﬁrm” an arbitrary
elements of W is. This concept will be used over and over throughout the
paper. See, in particular, Deﬁnition 3.9, Theorem 3.13, Theorem 4.4 and
Proposition 4.7. Our main result in this section is Theorem 2.18, showing
that the ﬁrmness of elements cannot drop below a certain value once they
become suﬃciently long.
Definition 2.10. Let w ∈ W be represented by some reduced word w =
s1 · · · sℓ ∈ MS .
(i) We say that w is firm if i ≻w ℓ for all i ∈ {1, . . . , ℓ−1}. In other words,
w is ﬁrm if its ﬁnal letter sℓ is in the ﬁnal position in each possible
reduced representation of w. Equivalently, w is ﬁrm if and only if there
is a unique r ∈ S such that l(wr) < l(w).
(ii) Let F # (w) be the largest k such that w can be represented by a reduced
word in the form
s1 · · · sk tk+1 · · · tℓ , with s1 · · · sk ﬁrm.
We call F # (w) the firmness of w. We will also use the notation
F # (w) := F # (w).
Lemma 2.11. Let w = s1 · · · sk tk+1 · · · tℓ be a reduced word such that s1 · · · sk
is firm and F # (w) = k. Then
(i) |sk ti | = 2 for all i ∈ {k + 1, . . . , ℓ}.
(ii) i ≻w k for all i ∈ {1, . . . , k − 1}.
(iii) Let r ∈ S. If l(wr) > l(w), then F # (wr) ≥ F # (w).
Proof. (i) Assume the contrary and let j be minimal such that |sk tj | = ∞.
Using elementary operations to swap tj to the left in w as much as
possible, we rewrite
w ∼ s1 · · · sk t′1 · · · t′p tj · · ·
as a word with s1 · · · sk t′1 · · · t′p tj ﬁrm, which is a contradiction to the
maximality of k.
(ii) The fact that the preﬁx p = s1 · · · sk is ﬁrm tells us that i ≻p k for all
i ∈ {1, . . . , k − 1}. By Lemma 2.9, this implies that also i ≻w k for all
i ∈ {1, . . . , k − 1}.
(iii) Since l(wr) > l(w), ﬁrm preﬁxes of w are also ﬁrm preﬁxes of wr,
hence the result.
7

The following deﬁnition will be a useful tool to identify which letters of
the word appear in a ﬁrm subword.
Definition 2.12. Let w = s1 · · · sℓ ∈ MS be a reduced word and consider
the poset (Iw , ≺w ) as in Deﬁnition 2.7. For any i ∈ {1, . . . , ℓ}, we deﬁne
Iw (i) = j ∈ {1, . . . , ℓ} | j ≻w i .


In words, Iw (i) is the set of indices j such that sj comes at the left of si in
any reduced representation of the element w ∈ W .
Observation 2.13. Let w = s1 · · · sℓ ∈ MS be a reduced word.
(i) Let i ∈ {1, . . . , ℓ} and write Iw (i) = {j1 , . . . , jk } with jp < jp+1 for
all p. Then we can perform elementary operations on w so that
w ∼ s j 1 · · · s j k s i t1 · · · tq
and the word sj1 · · · sjk si is ﬁrm.
In particular, if Iw (i) = ∅, then we can rewrite w as si w1 .
(ii) If i ≻w j, then Iw (i) ( Iw (j).
(iii) It follows from (i) that F # (w) = maxi∈{1,...,ℓ} |Iw (i)| + 1.
Remark 2.14. If the Coxeter system (W, S) is spherical, then F # (w) = 1
for all w ∈ W . Indeed, as each pair of distinct generators commute, we
always have Iw (i) = ∅.
The next deﬁnition will allow us to deal with possibly inﬁnite words.
Definition 2.15. (i) A (ﬁnite or inﬁnite) sequence (r1 , r2 , . . . ) of letters in S will be called a reduced increasing sequence if l(r1 · · · ri ) <
l(r1 · · · ri ri+1 ) for all i ≥ 1.
(ii) Let w ∈ MS . A sequence (r1 , r2 , . . . ) of letters in S will be called a
reduced increasing w-sequence if l(wr1 · · · ri ) < l(wr1 · · · ri ri+1 ) for all
i ≥ 0.
Lemma 2.16. Let α = (r1 , r2 , . . . ) be a reduced increasing sequence in S.
Assume that each subsequence of α of the form
(ra1 , ra2 , . . . ) with |rai rai+1 | = ∞ for all i
has ≤ b elements. Then there is some positive integer f (b) depending only
on b and on the Coxeter system (W, S), such that α has ≤ f (b) elements.

8

Proof. We will prove this result by induction on |S|; the case |S| = 1 is
trivial.
Suppose now that |S| ≥ 2. If (W, S) is a spherical Coxeter group, then
the result is obvious since the length of any reduced increasing sequence is
bounded by the length of the longest element of W . We may thus assume
that there is some s ∈ S that does not commute with some other generator
in S \ {s}.
Since the sequence α is a reduced increasing sequence, we know that
between any two s’s, there must be some ti such that |sti | = ∞. Consider
the subsequence of α given by
(s, t1 , s, t2 , . . . ).
This subsequence has ≤ b elements by assumption, and between any two
generators s in the original sequence α, we only use letters in S \ {s}. The
result now follows from the induction hypothesis.
Lemma 2.17. Let w ∈ W . Then there is some k(w) ∈ N, depending only
on w, such that for every reduced increasing w-sequence (r1 , r2 , . . . ) in S,
we have
F # (wr1 · · · rk(w) ) > F # (w).
Proof. Assume that there is a reduced increasing w-sequence α = (r1 , r2 , . . . )
in S such that
F # (wr1 · · · ri ) = F # (w) for all i.
(∗)
Let w0 = w, wi = wi−1 ri and denote Ii = Iwi (i) for all i. Let b = F # (w).
By assumption (∗) and Observation 2.13(iii), we have |Ii | ≤ b − 1 for all
i. Moreover, if i < j with |ri rj | = ∞, then Ii ( Ij by Observations 2.8(i)
and 2.13(ii); it follows that each subsequence of α of the form
(ra1 , ra2 , . . . ) with |rai rai+1 | = ∞ for all i
has at most b elements. By Lemma 2.16, this implies that the sequence
α has at most f (b) elements. We conclude that every reduced increasing
w-sequence (r1 , r2 , . . . , rk(w) ) in S with k(w) := f (F # (w)) + 1 must have
strictly increasing ﬁrmness.
Theorem 2.18. Let (W, S) be a right-angled Coxeter system. For all n ≥ 0,
there is some d(n) ∈ N depending only on n, such that F # (w) > n for all
w ∈ W with l(w) > d(n).
Proof. This follows by induction on n from Lemma 2.17 since there are only
ﬁnitely many elements in W of any given length.

9

3

Right-angled buildings

We will start by recalling the procedure of “closing squares” in right-angled
buildings from [DMSS16] and we deﬁne the square closure of a set of chambers. Our goal in this section is to describe the square closure of a ball in the
building and to show that this is a bounded set, i.e., it has ﬁnite diameter;
see Theorem 3.13.

3.1

Preliminaries

Throughout this section, let (W, S) be a right-angled Coxeter system with
Coxeter diagram Σ and let ∆ be a right-angled building of type (W, S). We
regard buildings as chamber systems, following the notation in [Wei09].
Definition 3.1. Let δ : ∆×∆ → W be the Weyl distance of the building ∆.
The gallery distance between the chambers c1 and c2 is deﬁned as
dW (c1 , c2 ) := l(δ(c1 , c2 )),
i.e., the length of a minimal gallery between the chambers c1 and c2 .
For a ﬁxed chamber c0 ∈ Ch(∆) we deﬁne the spheres at a ﬁxed gallery
distance from c0 as
S(c0 , n) := {c ∈ Ch(∆) | dW (c0 , c) = n}
and the balls as
B(c0 , n) := {c ∈ Ch(∆) | dW (c0 , c) ≤ n}.
Definition 3.2. (i) Let c be a chamber in ∆ and R be a residue in ∆.
The projection of c on R is the unique chamber in R that is closest to
c and it is denoted by projR (c).
(ii) If R1 and R2 are two residues, then the set of chambers
projR1 (R2 ) := {projR1 (c) | c ∈ Ch(R2 )}
is again a residue and the rank of projR1 (R2 ) is bounded above by the
ranks of both R1 and R2 ; see [Cap14, Section 2].
(iii) The residues R1 and R2 are called parallel if projR1 (R2 ) = R1 and
projR2 (R1 ) = R2 .
In particular, if P1 and P2 are two parallel panels, then the chamber sets
of P1 and P2 are mutually in bijection under the respective projection maps
(see again [Cap14, Section 2]).

10

Definition 3.3. Let J ⊆ S. We deﬁne the set
J ⊥ = {t ∈ S \ J | ts = st for all s ∈ J}.
If J = {s}, then we write the set J ⊥ as s⊥ .
Proposition 3.4 ([Cap14, Proposition 2.8]). Let ∆ be a right-angled building of type (W, S).
(i) Any two parallel residues have the same type.
(ii) Let J ⊆ S. Given a residue R of type J, a residue R′ is parallel to R
if and only if R′ is of type J, and R and R′ are both contained in a
common residue of type J ∪ J ⊥ .
Proposition 3.5 ([Cap14, Corollary 2.9]). Let ∆ be a right-angled building.
Parallelism of residues of ∆ is an equivalence relation.
Another very important notion in right-angled buildings is that of wings,
introduced in [Cap14, Section 3]. For our purposes, it will be suﬃcient to
consider wings with respect to panels.
Definition 3.6. Let c ∈ Ch(∆) and s ∈ S. Denote the unique s-panel
containing c by Ps,c . Then the set of chambers
Xs (c) = {x ∈ Ch(∆) | projPs,c (x) = c}
is called the s-wing of c.
Notice that if P is any s-panel, then the set of s-wings of each of the
diﬀerent chambers of P forms a partition of Ch(∆) into equally many combinatorially convex subsets (see [Cap14, Proposition 3.2]).

3.2

Sets of chambers closed under squares

We start by presenting two results proved in [DMSS16, Lemmas 2.9 and 2.10]
that can be used in right-angled buildings to modify minimal galleries using
the commutation relations of the Coxeter group. We will refer to these
results as the “Closing Squares Lemmas” (see also Figure 1 below). We
s
use the notation c1 ∼ c2 to denote that two chambers c1 and c2 of ∆ are
s-adjacent, i.e., are contained in a common s-panel of ∆.
Lemma 3.7 (Closing Squares 1). Let c0 be a fixed chamber in ∆. Let
c1 , c2 ∈ S(c0 , n) and c3 ∈ S(c0 , n + 1) such that
t

c1 ∼ c3

and

s

c2 ∼ c3

for some s 6= t. Then |st| = 2 in Σ and there exists c4 ∈ S(c0 , n − 1) such
that
s
t
c1 ∼ c4 and c2 ∼ c4 .
11

Lemma 3.8 (Closing Squares 2). Let c0 be a fixed chamber in ∆. Let
c1 , c2 ∈ S(c0 , n) and c3 ∈ S(c0 , n − 1) such that
s

c1 ∼ c2

and

t

c2 ∼ c3

for some s 6= t. Then |st| = 2 in Σ and there exists c4 ∈ S(c0 , n − 1) such
that
t
s
c1 ∼ c4 and c3 ∼ c4 .
c3
t

n+1

c1

s

c1

c2
s

n

n
t

t

t

c4
n−1

c4
..
.

c2

s

s
..
.

c3

n−1

c0
c0

(a) Lemma 3.7

(b) Lemma 3.8

Figure 1: Closing Squares Lemmas
Definition 3.9. Let c0 be a ﬁxed chamber of ∆ and let n ∈ N.
(i) Let c ∈ Ch(∆). Then we call c firm with respect to c0 if and only if
δ(c0 , c) ∈ W is ﬁrm (as in Deﬁnition 2.10(i)).
(ii) We will create a partition of the sphere S(c0 , n) by deﬁning
A1 (n) = {c ∈ S(c0 , n) | c is ﬁrm},
A2 (n) = {c ∈ S(c0 , n) | c is not ﬁrm},
as in Figure 2. Notice that this is equivalent to the deﬁnition given in
[DMSS16, Deﬁnition 4.3].
(iii) Let c ∈ S(c0 , k) for some k > n. We say that c is n-flexible with respect
to c0 if for each minimal gallery γ = (c0 , c1 , . . . , cn+1 , . . . , ck = c) from
c0 to c, none of the chambers cn+1 , . . . , ck is ﬁrm. By convention, all
chambers of B(c0 , n) are also n-ﬂexible with respect to c0 .
Observe that a chamber c is n-ﬂexible with respect to c0 if and only
if F # (δ(c0 , c)) ≤ n. In particular, if c is n-ﬂexible, then so is any
chamber on any minimal gallery between c0 and c.
(iv) We deﬁne the n-flex of c0 , denoted by Flex(c0 , n), to be the set of all
chambers of ∆ that are n-ﬂexible with respect to c0 .

12

n+1
t

t

t

t

c1

c2

t

c2

c1

t
n

c3

t s

s s s

s t

c3
s

n

t
n−1

n−1
..
.

..
.
c0

c0

(a) ci ﬁrm: for all t 6= s,
l(δ(c0 , ci )t) > l(δ(c0 , ci )).

(b) ci not ﬁrm: for some t 6= s,
l(δ(c0 , ci )t) < l(δ(c0 , ci )).

Figure 2: Partition of S(c0 , n).
We also record the following result, which we rephrased in terms of ﬁrm
chambers; its Corollary 3.11 will be used several times in Section 4.
Lemma 3.10 ([DMSS16, Lemma 2.15]). Let c0 be a fixed chamber of ∆
and let s ∈ S. Let d ∈ S(c0 , n) and e ∈ B(c0 , n + 1) \ Ch(Ps,d ). If c :=
projPs,d (e) ∈ S(c0 , n + 1), then c is not firm with respect to c0 .
Corollary 3.11. Let c0 ∈ Ch(∆) and c ∈ S(c0 , n + 1) such that c is firm
with respect to c0 . Let d be the unique chamber of S(c0 , n) adjacent to c and
let s = δ(d, c) ∈ S. Then B(c0 , n) ⊂ Xs (d).
Proof. Let e ∈ B(c0 , n). If e = d, then of course e ∈ Xs (d), so assume e 6= d;
then e ∈ B(c0 , n + 1) \ Ch(Ps,d ). Notice that all chambers of Ps,d \ {d} have
the same Weyl distance from c0 as c and hence are ﬁrm. By Lemma 3.10, this
implies that the projection of e on Ps,d must be equal to d, so by deﬁnition
of the s-wing Xs (d), we get e ∈ Xs (d).
We now come to the concept of the square closure of a set of chambers
of ∆.
Definition 3.12. (i) We say that a subset T ⊆ W is closed under squares
if the following holds:
If wsi and wsj are contained in T for some w ∈ T with
|si sj | = 2, si 6= sj and l(wsi ) = l(wsj ) = l(w) + 1, then also
wsi sj = wsj si is an element of T .
(ii) Let c0 be a ﬁxed chamber of ∆. A set of chambers C ⊆ Ch(∆) is
closed under squares with respect to c0 if for each n ∈ N, the following
holds (see Figure 1a):
13

If c1 , c2 ∈ C ∩ S(c0 , n) and c4 ∈ C ∩ S(c0 , n − 1) such that
sj
s
c4 ∼i c1 and c4 ∼ c2 for some |si sj | = 2 with si 6= sj , then
sj
the unique chamber c3 ∈ S(c0 , n + 1) such that c3 ∼ c1 and
s
c3 ∼i c2 is also in C .
In particular, if C is closed under squares with respect to c0 , then the
set of Weyl distances {δ(c0 , c) | c ∈ C } ⊆ W is closed under squares.
(iii) Let c0 ∈ Ch(∆) and let C ⊆ Ch(∆). We deﬁne the square closure of C
with respect to c0 to be the smallest subset of Ch(∆) containing C
and closed under squares with respect to c0 .
Theorem 3.13. Let c0 ∈ Ch(∆) and let n ∈ N. The square closure of
B(c0 , n) with respect to c0 is Flex(c0 , n). Moreover, the set Flex(c0 , n) is
bounded.
Proof. We will ﬁrst show that Flex(c0 , n) is indeed closed under squares. Let
c3 be a chamber in Flex(c0 , n) at Weyl distance w from c0 and let c1 and c2
be chambers in Flex(c0 , n) adjacent to c3 , at Weyl distance wsi and wsj from
c0 , respectively, such that |si sj | = 2 and l(wsi ) = l(wsj ) = l(w) + 1. Let c3
be the unique chamber at Weyl distance wsi sj from c0 that is sj -adjacent
to c1 and si -adjacent to c2 .
c3 = vk
si

sj

sk

vk−1

c2
si
sk

c1

sj

d2

si

sj

l(w) + 1

sk

vk−2
d1

l(w)

vk−3
l(w) − 1
vn+1
n

vn
c0 = v0

Figure 3: Proof of Theorem 3.13
Our aim is to show that also c3 is an element of Flex(c0 , n). If l(wsi sj ) ≤ n,
then this is obvious, so we may assume that l(wsi sj ) > n.

14

Let γ = (c0 = v0 , . . . , vn+1 , . . . , vk = c3 ) be an arbitrary minimal gallery
between c0 and c3 , as in Figure 3 (so k = l(w) + 2 > n). We have to show
that none of the chambers vn+1 , . . . , vk is ﬁrm with respect to c0 . This is
clear for vk = c3 .
If k = n + 1, then there is nothing left to show, so assume k ≥ n + 2. If
vk−1 ∈ {c1 , c2 }, then vk−1 is n-ﬂexible by assumption, and since k − 1 > n
it is not ﬁrm. (In fact, this shows immediately that in this case, none of the
chambers vn+1 , . . . , vk−1 is ﬁrm). So assume that vk−1 is distinct from c1
and c2 ; then vk−1 is sk -adjacent to c3 for some sk diﬀerent from si and sj .
Then by closing squares (Lemma 3.7), we have |sj sk | = 2 and there is a
sj
s
chamber d1 ∈ S(c0 , l(w)) such that d1 ∼ vk−1 and d1 ∼k c1 . Similarly, there
s
s
is a chamber d2 ∈ S(c0 , l(w)) such that d2 ∼i vk−1 and d2 ∼k c2 . Hence vk−1
is not ﬁrm with respect to c0 .
Continuing this argument inductively (see Figure 3), we conclude that
none of the chambers vn+1 , . . . , vk is ﬁrm with respect to c0 . Hence c3 is
n-ﬂexible; we conclude that Flex(c0 , n) is closed under squares with respect
to c0 .
Conversely, let C be a set of chambers closed under squares that contains
B(c0 , n); we have to prove that Flex(c0 , n) ⊆ C . So let c ∈ Flex(c0 , n) be
arbitrary; we will show by induction on k := dW (c0 , c) that c ∈ C . This is
obvious for k ≤ n, so assume k > n. Then c is not ﬁrm, hence there exist
s
s
c1 , c2 ∈ S(c0 , k − 1) such that c1 ∼1 c and c2 ∼2 c for some s1 6= s2 ∈ S. By
s
Lemma 3.7 we have |s1 s2 | = 2 and there is d ∈ S(c0 , k − 2) such that d ∼2 c1
s1
and d ∼ c2 .
Since c is n-ﬂexible and c1 , c2 and d all lie on some minimal gallery
between c0 and c, it follows that also c1 , c2 and d are n-ﬂexible. By the
induction hypothesis, all three elements are contained in C . Since C is
assumed to be closed under squares, however, we immediately deduce that
also c ∈ C .
We conclude that Flex(c0 , n) is the square closure of B(c0 , n) with respect
to c0 .
We ﬁnally show that Flex(c0 , n) is a bounded set. Recall that a chamber c
is contained in Flex(c0 , n) if and only if F # (δ(c0 , c)) ≤ n. By Theorem 2.18,
there is a constant d(n) such that F # (w) > n for all w ∈ W with l(w) >
d(n). This shows that Flex(c0 , n) ⊆ B(c0 , d(n)) is indeed bounded.

4

The automorphism group of a right-angled building

In this section, we study the group Aut(∆) of type-preserving automorphisms of a thick semi-regular right-angled building ∆. We will ﬁrst study
the action of a ball ﬁxator and introduce root wing groups. Next, we will
15

characterize the compact open subgroups of Aut(∆). Finally, when the
building is locally ﬁnite, we will show that any proper open subgroup of
Aut(∆) is a ﬁnite index subgroup of the stabilizer of a proper residue; see
Theorem 4.29.
Definition 4.1. Let ∆ be a right-angled building of type (W, S). Then ∆ is
called semi-regular if for each s, all s-panels of ∆ have the same number qs
of chambers. In this case, the building is said to have prescribed thickness
(qs )s∈S in its panels.
By [HP03, Proposition 1.2], there is a unique right-angled building of
type (W, S) of prescribed thickness (qs )s∈S for any choice of cardinal numbers qs ≥ 1.
Theorem 4.2 ([KT12, Theorem B], [Cap14, Theorem 1.1]). Let ∆ be a
thick semi-regular building of right-angled type (W, S). Assume that (W, S)
is irreducible and non-spherical. Then the group Aut(∆) of type-preserving
automorphisms of ∆ is abstractly simple and acts strongly transitively on ∆.
The strong transitivity has ﬁrst been shown by Angela Kubena and Anne
Thomas [KT12] and has been reproved by Pierre-Emmanuel Caprace in the
same paper where he proved the simplicity [Cap14]. In our proof of Proposition 4.7 below, we will adapt Caprace’s proof of the strong transitivity to
a more speciﬁc setting.
The following extension result is very powerful and will be used in the
proof of Theorem 4.4 below.
Proposition 4.3 ([Cap14, Proposition 4.2]). Let ∆ be a semi-regular rightangled building. Let s ∈ S and P be an s-panel. Given any permutation
θ ∈ Sym(Ch(P)), there is some g ∈ Aut(∆) stabilizing P satisfying the
following two conditions:
(a) g|Ch(P) = θ;
(b) g fixes all chambers of ∆ whose projection on P is fixed by θ.

4.1

The action of the fixator of a ball in ∆

In this section we study the action of the ﬁxator K in Aut(∆) of a ball
B(c0 , n) of radius n around a chamber c0 . Our goal will be to prove that
the ﬁxed point set ∆K coincides with the square closure of the ball B(c0 , n)
with respect to c0 , which is Flex(c0 , n), and which we know is bounded by
Theorem 3.13.
Theorem 4.4. Let ∆ be a thick semi-regular right-angled building. Let c0
be a fixed chamber of ∆ and let n ∈ N. Consider the pointwise stabilizer
K = FixAut(∆) (B(c0 , n)) in Aut(∆) of the ball B(c0 , n).
Then the fixed-point set ∆K is equal to the bounded set Flex(c0 , n).
16

Proof. Recall from Theorem 3.13 that Flex(c0 , n) is precisely the square closure of B(c0 , n) with respect to c0 . First, notice that the ﬁxed point set of
any automorphism ﬁxing c0 is square closed with respect to c0 because the
chamber “closing the square” is unique (see Deﬁnition 3.12(ii)). It immediately follows that Flex(c0 , n) ⊆ ∆K .
We will now show that if c is a chamber not in Flex(c0 , n), then there
exists a g ∈ K not ﬁxing c. Since c is not n-ﬂexible, there exists a chamber
d on some minimal gallery between c0 and c with k := dW (c0 , d) > n such
that d is ﬁrm. Notice that any automorphism ﬁxing c0 and c ﬁxes every
chamber on any minimal gallery between c0 and c, so it suﬃces to show
that there exists a g ∈ K not ﬁxing d.
s
Since d is ﬁrm, there is a unique chamber e ∈ S(c0 , k − 1) such that e ∼ d
for some s ∈ S. By Corollary 3.11, B(c0 , n) ⊆ Xs (e), where Xs (e) is the
s-wing of ∆ corresponding to e.
Now take any permutation θ of Ps,e ﬁxing e and mapping d to some third
chamber d′′ diﬀerent from d and e (which exists because ∆ is thick). By
Proposition 4.3, there is an element g ∈ Aut(∆) ﬁxing Xs (e) and mapping
d to d′′ . In particular, g belongs to K and does not ﬁx d, as required.
We conclude that ∆K = Flex(c0 , n). The fact that this set is bounded
was shown in Theorem 3.13.

4.2

Root wing groups

In this section we deﬁne groups that resemble root groups, using the partition of the chambers of a right-angled building by wings; we call these
groups root wing groups.
We show that a root wing group acts transitively on the set of apartments
of ∆ containing the given root. We also prove that the root wing groups
corresponding to roots disjoint from a ball B(c0 , n) are contained in the
ﬁxator of that ball in the automorphism group.
We ﬁrst ﬁx some notation for the rest of this section.
Notation 4.5. (i) Fix a chamber c0 ∈ Ch(∆) and an apartment A0 containing c0 (which can be considered as the fundamental chamber and
the fundamental apartment). Let Φ denote the set of roots of A0 . For
each α ∈ Φ, we write −α for the root opposite α in A0 .
(ii) We will write A0 for the set of all apartments containing c0 . For any
A ∈ A0 , we will denote its set of roots by ΦA .
(iii) For any k ∈ N, we write Kr := FixAut(∆) (B(c0 , r)).
Definition 4.6. (i) When α ∈ ΦA is a root in an apartment A, its wall
∂α consists of the panels of ∆ having chambers in both α and −α.
Since the building is right-angled, these panels all have the same type
17

s ∈ S, which we refer to as the type of α and write as type(α) = s.
Notice that the s-wings of A are precisely the roots of A of type s.
(ii) Let α ∈ ΦA of type s and let c ∈ α be such that Ps,c ∈ ∂α. Then we
deﬁne the root wing group Uα as
Uα := Us (c) := FixAut(∆) (Xs (c)).
Observe that Uα does not depend of the choice of the chamber c as
all panels in the wall ∂α are parallel (see Deﬁnition 3.2(iii)) and hence
determine the same s-wings in ∆.
The fact that these groups behave, to some extent, like root groups in
Moufang spherical buildings or Moufang twin buildings, is illustrated by the
following fact.
Proposition 4.7. Let α ∈ ΦA be a root. The root wing group Uα acts
transitively on the set of apartments of ∆ containing α.
Proof. We carefully adapt the proof of the strong transitivity of Aut(∆) from
[Cap14, Proposition 6.1]. Let c be a chamber of α on the boundary and let
A and A′ be two apartments of ∆ containing α. The strategy in loc. cit.
(where A and A′ are arbitrary apartments containing c) is to construct an
inﬁnite sequence of automorphisms g0 , g1 , g2 , . . . such that
(a) each gn ﬁxes the ball B(c, n − 1) pointwise;
(b) let An := gn gn−1 · · · g0 (A); then An ∩ A′ ⊇ B(c, n) ∩ A′ .
We will show that the elements gi constructed in loc. cit. are all contained
in Uα ; the result then follows because Uα is a closed subgroup of Aut(∆).
To construct the element gn+1 , we consider the set E of chambers in
B(c, n + 1) ∩ A′ that are not contained in An (as in loc. cit.). The crucial
observation now is that by Theorem 4.4, the chambers of E are ﬁrm with
respect to c. Hence, for each x ∈ E, there is a unique chamber y ∈ S(c, n)
that is s-adjacent to x (for some s ∈ S). The element gn+1 constructed in
loc. cit. is then contained in the group generated by the subgroups Us (y)
for such pairs (y, s) corresponding to the various elements of E. However,
because the elements of E are ﬁrm, the root α is contained in each root
corresponding to a pair (y, s) in A′ ; [Cap14, Lemma 3.4(b)] now implies
that each such group Us (y) is contained in Uα .
Remark 4.8. The group Uα does not, in general, act sharply transitively
on the set of apartments containing α. This is clear already in the case of
trees: an automorphism ﬁxing a half-tree and an apartment need not be
trivial.
Corollary 4.9. Let α ∈ ΦA be a root of type s and let c, c′ be two s-adjacent
chambers of A with c ∈ α and c′ ∈ −α. Then there exists an element in
hUα , U−α i stabilizing A and interchanging c and c′ .
18

Proof. Let A′ be an apartment diﬀerent from A containing α (which exists
because ∆ is thick) and let β be the root opposite α in A′ . By Proposition 4.7, there is some g ∈ Uα mapping −α to β. Similarly, there is some
h ∈ U−α mapping β to α. Let γ := h.α; then there exists a third automorphism g′ ∈ Uα mapping γ to −α. The composition g′ hg ∈ Uα U−α Uα is the
required automorphism.
Next we present a property similar to the FPRS (“Fixed Points of Root
Subgroups”) property introduced in [CR09] for groups with a twin root
datum. It is the analogous statement of [CM13, Lemma 3.8], but in the case
of right-angled buildings, we can be more explicit.
Lemma 4.10. For every root α ∈ Φ with dist(c0 , α) > r, the group U−α is
contained in Kr = FixAut(∆) (B(c0 , r)).
Proof. Let α be a root at distance n > r from c0 and let s be the type of α.
Let c = projα (c0 ) and let c′ be the other chamber in Ps,c ∩ A0 ; notice that
c′ ∈ S(c0 , n − 1). We will show that B(c0 , r) ⊆ Xs (c′ ), which will then of
course imply that U−α = Us (c′ ) ⊆ Kr .
The chamber c is ﬁrm with respect to c0 because if c would be t-adjacent
to some chamber at distance n − 1 from c0 for some t 6= s, then ∂α would
contain panels of type s and of type t, which is impossible. Corollary 3.11
now implies that B(c0 , n−1) ⊆ Xs (c′ ), so in particular B(c0 , r) ⊆ Xs (c′ ).
Following the idea of [CM13, Lemmas 3.9 and 3.10], we present two
variations on the previous lemma that allow us to transfer the results to
other apartments containing the chamber c0 .
Lemma 4.11. Let g ∈ Aut(∆) and let A ∈ A0 containing the chamber
d = gc0 . Let b ∈ StabAut(∆) (c0 ) such that A = bA0 , and let α = bα0 be a
root of A with α0 ∈ Φ.
If dist(d, −α) > r, then bUα0 b−1 ⊆ gKr g−1 .
Proof. Analogous to the proof of [CM13, Lemma 3.9].
Definition 4.12 ([CM13, Section 2.4]). Let w ∈ W .
(i) A root α ∈ Φ is called w-essential if wn α ( α for some n ∈ Z.
(ii) A wall is called w-essential if it is the wall ∂α of some w-essential
root α.
Lemma 4.13. Let A ∈ A0 and let b ∈ StabAut(∆) (c0 ) such that A =
bA0 . Also, let α = bα0 (with α0 ∈ Φ) be a w-essential root for some
w ∈ StabAut(∆) (A)/ Fix Aut(∆) (A). Let g ∈ StabAut(∆) (A) be a representative
of w.

19

Then there exists some n ∈ Z such that
Uα0 ⊆ b−1 gn Kr g−n b
U−α0 ⊆ b

−1 −n

g

and

n

Kr g b.

Proof. The proof can be copied ad verbum from [CM13, Lemma 3.10].

4.3

Compact open subgroups of Aut(∆)

We now focus on the description of open subgroups of the automorphism
group of ∆. The main result of the next section will be that any proper
open subgroup of the automorphism group of a locally ﬁnite thick semiregular right-angled building ∆ is contained with ﬁnite index in the setwise
stabilizer in Aut(∆) of a proper residue of ∆ (see Theorem 4.29 below).
We will split the proof in the cases where the open subgroup is compact
and non-compact. In this section, we ﬁrst deal with the (easier) compact
case.
Throughout this section, we assume that ∆ is a thick irreducible semiregular right-angled building (not necessarily locally ﬁnite) and we will denote the Davis realization of ∆ by X (see [Dav98]). Using the work developed in Section 4.1, we can prove that an open subgroup of Aut(∆) which
is locally X-elliptic on X must be compact.
Definition 4.14. A group acting continuously on a space X is called locally
X-elliptic if every compactly generated subgroup of Aut(∆) ﬁxes a point
in X.
Proposition 4.15. Let H be an open subgroup of Aut(∆). Then the following are equivalent:
(a) H is locally X-elliptic;
(b) H fixes a point of X;
(c) H is a finite index subgroup of the stabilizer of a spherical residue of ∆;
(d) H is compact.
Proof. Notice that the points of X correspond precisely to the spherical
residues of ∆ and that the maximal compact open subgroups of Aut(∆)
are precisely the stabilizers of a maximal spherical residue, so the only nontrivial implication is (a) =⇒ (b).
So assume that H is locally X-elliptic. We will rely on [CL10, Theorem 1.1] to show ﬁrst that H has a global ﬁxed point on X or H ﬁxes an
end of X. Notice that X has ﬁnite geometric dimension (namely equal to
the highest possible rank of a spherical parabolic subgroup of (W, S)) and
hence also ﬁnite telescopic dimension (see loc. cit. for these notions). For
20

each ﬁnite subset F ⊂ H, we let XF be the set of ﬁxed points in X of hF i;
then each XF is non-empty because X is locally X-elliptic, and the collection {XF } is a ﬁltering family of closed convex subspaces of X. By [CL10,
T
Theorem 1.1], either the intersection XF is nonempty, or the intersection
T
of the boundaries ∂XF is nonempty. In the ﬁrst case, H ﬁxes a point
of X; in the second case, H ﬁxes an end of X.
Assume that H ﬁxes an end of X; we will show that H then also ﬁxes
a point of X. Since H is open, it contains the ﬁxator of some ﬁnite ball,
i.e., K := FixAut(∆) (B(c0 , n)) ⊆ H for some c0 ∈ Ch(∆) and some n ∈ N.
Moreover, for each h ∈ H, the group Hh := hh, Ki is open and compactly
generated. Since H is locally X-elliptic by assumption, each Hh has a global
ﬁxed point, i.e., X Hh 6= ∅.
S
Hence H = Hh with each Hh open and compactly generated and we
can take this union to be countable because Aut(∆) is second countable.
Observe that X Hh ⊆ X K for each h ∈ H. By Theorem 4.4, the ﬁxed-point
set X K is bounded. Since a countable intersection of compact bounded nonempty sets is non-empty, we conclude that X H is non-empty; hence H ﬁxes
a point of X, as claimed.

4.4

Open subgroups of Aut(∆), with ∆ locally finite

We will assume from now on that ∆ is a thick irreducible semi-regular locally
finite right-angled building. Consider an open subgroup H of Aut(∆) and
assume that H is non-compact.
Definition 4.16. We continue to use the conventions from Notation 4.5
and we will identify the apartment A0 with W .
(i) Given a root α ∈ Φ, let rα denote the unique reﬂection of W setwise
stabilizing the panels in ∂α and let Uα be the root wing group introduced in Deﬁnition 4.6. By Corollary 4.9, the reﬂection rα ∈ W lifts
to an automorphism nα ∈ hUα , U−α i ≤ Aut(∆) stabilizing A0 .
(ii) For each c ∈ Ch(∆) and each subset J ⊆ S, we write RJ,c for the
residue of ∆ of type J containing c. We use the shorter notation RJ :=
RJ,c0 when c = c0 . Moreover, we write PJ := StabAut(∆) (RJ ), and we
call this a standard parabolic subgroup of Aut(∆). Any conjugate of PJ ,
i.e., any stabilizer of an arbitrary residue, is then called a parabolic
subgroup.
(iii) Let J ⊆ S be minimal such that there is a g ∈ Aut(∆) such that
H ∩ g−1 PJ g has ﬁnite index in H. In particular, J is essential (see
Deﬁnition 2.2(iii)). See also [CM13, Lemma 3.4].
For such a g, we set H1 = gHg−1 ∩ PJ . Thus H1 stabilizes RJ and it
is an open subgroup of Aut(∆) contained in gHg−1 with ﬁnite index;
21

since H is non-compact, so is H1 . Hence we may assume without loss
of generality that g = 1 and hence H1 = H ∩ PJ has ﬁnite index in H.
(iv) Let A0 be the set of apartments of ∆ containing c0 . For A ∈ A0 we let
NA := StabH1 (A)

and WA := NA / FixH1 (A),

which we identify with a subgroup of W . For h ∈ NA , let h denote its
image in WA ≤ W .
The idea will be to prove that H1 contains a hyperbolic element h such
that the chamber c0 achieves the minimal displacement of h. Moreover, we
can ﬁnd the element h in the stabilizer in H1 of an apartment A1 containing c0 . Thus we can identify it with an element h of W and consider its
parabolic closure (see Deﬁnition 2.2(iv)). The key point will be to prove
that the type of Pc(h) is J, which will be achieved in Lemma 4.24.
We will also show that H1 acts transitively on the chambers of RJ ; this
will allow us to conclude that any open subgroup of Aut(∆) containing H1
as a ﬁnite index subgroup is contained in the stabilizer of RJ∪J ′ for some
spherical subset J ′ of J ⊥ (Proposition 4.26).
This strategy is analogous (and, of course, inspired by) [CM13, Section 3]. As the arguments of loc. cit. are of a geometric nature, we will
be able to adapt them to our setting. The root groups associated with the
Kac–Moody group in that paper can be replaced by the root wing groups
deﬁned in Section 4.2. It should not come as a surprise that many of our
proofs will simply consist of appropriate references to arguments in [CM13].
Lemma 4.17. For all A ∈ A0 , there exists a hyperbolic automorphism
h ∈ NA such that
Pc(h) = hrα | α is an h-essential root of Φi
and is of finite index in Pc(WA ).
Proof. Using the fact that the reﬂections rα lift to elements nα ∈ hUα , U−α i
(see Deﬁnition 4.16(i)), the proof is the same as for [CM13, Lemma 3.5].
Lemma 4.18. There exists an apartment A ∈ A0 such that the orbit NA .c0
is unbounded. In particular, the parabolic closure in W of WA is nonspherical.
Proof. The proofs of [CM13, Lemmas 3.6 and 3.7] continue to hold without
a single change. Notice that this depends crucially on the fact that H1 is
non-compact.
Definition 4.19. (i) Let A1 ∈ A0 be an apartment such that the essential component of Pc(WA1 ) is non-empty and maximal with respect
22

to this property (see Deﬁnition 2.2(iii)); such an apartment exists by
Lemma 4.18. Choose h1 ∈ NA1 as in Lemma 4.17. In particular, h1 is
a hyperbolic element of H1 .
(ii) Up to conjugating H1 by an element of StabAut(∆) (RJ ), we can assume
without loss of generality that Pc(h1 ) is a standard parabolic subgroup
that is non-spherical and has essential type I (6= ∅). Moreover, the type
I is maximal in the following sense: if A ∈ A0 is such that Pc(WA )
contains a parabolic subgroup of essential type IA with I ⊆ IA , then
I = IA .
Definition 4.20. Recall that Φ is the set of roots of the apartment A0 . For
each T ⊆ S, let
ΦT := {α ∈ Φ | RT contains at least one panel of ∂α}
and
L+
T := hUα | α ∈ ΦT i,
where Uα is the root wing group introduced in Deﬁnition 4.6.
Our next goal is to prove that H1 contains L+
J , where J is as in Deﬁnition 4.16(iii); as we will see in Lemma 4.22 below, this fact is equivalent to
H1 being transitive on the chambers of RJ .
We will need the results in Section 4.2 regarding ﬁxators of balls and
root wing groups.
Notation 4.21. Since H1 is open, we ﬁx, for the rest of the section, some
r ∈ N such that FixAut(∆) (B(c0 , r)) ⊆ H1 .
The next lemma corresponds to [CM13, Lemma 3.11], but some care is
needed because of our diﬀerent deﬁnition of the groups Uα .
Lemma 4.22. Let T ⊆ S be essential and let A ∈ A0 . Then the following
are equivalent:
(a) H1 contains L+
T;
(b) H1 is transitive on RT ;
(c) NA is transitive on RT ∩ A;
(d) WA contains the standard parabolic subgroup WT of W .
Proof. It is clear that (c) and (d) are equivalent.
We ﬁrst show that (a) implies (c). It suﬃces to show that for each
chamber c1 of A that is s-adjacent to c0 for some s ∈ T , there is an element
of NA mapping c0 to c1 . Let α be the root of A0 containing c0 but not
the chamber c2 in A0 that is s-adjacent to c0 ; notice that Uα and U−α
23

are contained in L+
T . By Proposition 4.7, there is some g ∈ Uα ﬁxing c0
and mapping c1 to c2 . Now the element nα ∈ hUα , U−α i stabilizes A0 and
interchanges c0 and c2 ; it follows that the conjugate g−1 nα g stabilizes A and
interchanges c0 and c1 , as required.
The proofs of the implications (d) ⇒ (b) ⇒ (a) are exactly as in [CM13,
Lemma 3.11].
The next statement is the analogue of [CM13, Lemma 3.12].
Lemma 4.23. Let A ∈ A0 . There exists IA ⊆ S such that WA contains a
parabolic subgroup PIA of W of type IA as a finite index subgroup.
Proof. The proof can be copied ad verbum from [CM13, Lemma 3.12].
For each A ∈ A0 , we ﬁx such an IA ⊆ S; without loss of generality,
we may assume that IA is essential. We also consider the corresponding
parabolic subgroup PIA contained in WA . Observe that PIA1 has ﬁnite index
in Pc(WA1 ) by Lemma 2.3, where A1 is as in Deﬁnition 4.19(i). Therefore
I = IA 1 .
The next task in the process of showing that H1 contains L+
J is to prove
that J = I, which is achieved by the following sequence of steps, each of
which follows from the previous ones and which are analogues of results in
[CM13].
Lemma 4.24. Let A ∈ A0 and let I and J be as in Definition 4.19(ii)
and 4.16(iii), respectively. Then:
(i) H1 contains L+
I ;
(ii) IA ⊂ I;
(iii) WA contains WI as a subgroup of finite index;
(iv) I = J.
Proof. (i) This follows from the fact that I = IA1 and PI = WI ; the
conclusion follows from Lemma 4.22.
(ii) See [CM13, Lemma 3.14].
(iii) See [CM13, Lemma 3.15].
(iv) See [CM13, Lemma 3.16].
Corollary 4.25. The group H1 acts transitively on the chambers of RJ .
Proof. This follows by combining Lemmas 4.22 and 4.24.

24

We are approaching our main result; the following proposition already
shows, in particular, that H is contained in the stabilizer of a residue, and it
will only require slightly more eﬀort to show that it is a finite index subgroup
of such a stabilizer.
Proposition 4.26. Every subgroup of Aut(∆) containing H1 as a subgroup
of finite index is contained in a stabilizer StabAut(∆) (RJ∪J ′ ), where J ′ is a
spherical subset of J ⊥ .
Proof. The proof is exactly the same as in [CM13, Lemma 3.19].
Notice that since ∆ is irreducible, the index set J ∪ J ′ is only equal to
S if already J = S.
Lemma 4.27. The group H1 is a finite index subgroup of StabAut(∆) (RJ ).
Proof. Let G := StabAut(∆) (RJ ). We already know that H1 stabilizes RJ
(see Deﬁnition 4.16(iii)) and acts transitively on the set of chambers of RJ
(see Corollary 4.25). Notice that the stabilizer in G of a chamber of RJ is
compact, hence H1 is a cocompact subgroup of G. Since H1 is also open
in G, we conclude that H1 is a ﬁnite index subgroup of G.
Lemma 4.28. For every spherical J ′ ⊆ J ⊥ , the index of StabAut(∆) (RJ ) in
StabAut(∆) (RJ∪J ′ ) is finite.
Proof. By [Cap14, Lemma 2.2], we have Ch(RJ∪J ′ ) = Ch(RJ ) × Ch(RJ ′ ).
As J ′ is spherical, the chamber set Ch(RJ ′ ) is ﬁnite; the result follows.
We are now ready to prove our main theorem.
Theorem 4.29. Let ∆ be a thick irreducible semi-regular locally finite rightangled building of rank at least 2. Then any proper open subgroup of Aut(∆)
is contained with finite index in the stabilizer in Aut(∆) of a proper residue.
Proof. Let H be a proper open subgroup of Aut(∆). If H is compact, then
the result follows from Proposition 4.15.
So assume that H is not compact. By Deﬁnition 4.16(iii), we may assume
that H contains a ﬁnite index subgroup H1 which, by Corollary 4.25, acts
transitively on the chambers of some residue RJ . By Proposition 4.26, H is
a subgroup of G := StabAut(∆) (RJ∪J ′ ) for some spherical J ′ ⊆ J ⊥ . On the
other hand, Lemmas 4.27 and 4.28 imply that H1 is a ﬁnite index subgroup
of G; since H1 is a ﬁnite index subgroup of H, it follows that also H has
ﬁnite index in G.
It only remains to show that RJ∪J ′ is a proper residue. If not, then
G = Aut(∆), but since G is simple (Theorem 4.2) and inﬁnite, it has no
proper ﬁnite index subgroups. Since H is a proper open subgroup of G, the
result follows.
25

5

Two applications of the main theorem

In this last section we present two consequences of Theorem 4.29, both of
which were suggested to us by Pierre-Emmanuel Caprace. The ﬁrst states
that the automorphism group of a locally ﬁnite thick semi-regular rightangled building ∆ is Noetherian (see Deﬁnition 5.1); the second deals with
reduced envelopes in Aut(∆).
Definition 5.1. We call a topological group Noetherian if it satisﬁes the
ascending chain condition on open subgroups.
We will prove that the group Aut(∆) is Noetherian by making use of the
following characterization.
Lemma 5.2 ([CM13, Lemma 3.22]). Let G be a locally compact group.
Then G is Noetherian if and only if every open subgroup of G is compactly
generated.
Proposition 5.3. Let ∆ be a locally finite thick semi-regular right-angled
building. Then the group Aut(∆) is Noetherian.
Proof. By Lemma 5.2, we have to show that every open subgroup of Aut(∆)
is compactly generated. By Theorem 4.29, every open subgroup of Aut(∆)
is contained with ﬁnite index in the stabilizer of a residue of ∆.
Stabilizers of residues are compactly generated, since they are generated
by the stabilizer of a chamber c0 (which is a compact open subgroup) together with a choice of elements mapping c0 to each of its (ﬁnitely many)
neighbors. Since a closed cocompact subgroup of a compactly generated
group is itself compactly generated (see [MS59]), we conclude that indeed
every open subgroup of Aut(∆) is compactly generated and hence Aut(∆)
is Noetherian.
Our next application deals with reduced envelopes, a notion introduced
by Colin Reid in [Rei16b] in the context of arbitrary totally disconnected
locally compact (t.d.l.c.) groups.
Definition 5.4. (i) Two subgroups H1 and H2 of a group G are called
commensurable if H1 ∩ H2 has ﬁnite index in both H1 and H2 .
(ii) Let G be a totally disconnected locally compact (t.d.l.c.) group and
let H ≤ G be a subgroup. An envelope of H in G is an open subgroup
of G containing H. An envelope E of H is called reduced if for any
open subgroup E2 with [H : H ∩ E2 ] < ∞ we have [E : E ∩ E2 ] < ∞.
Not every subgroup of G has a reduced envelope, but clearly any two
reduced envelopes of a given group are commensurable.

26

Theorem 5.5 ([Rei16a, Theorem B]). Let G be a t.d.l.c. group and let H
be a (not necessarily closed) compactly generated subgroup of G. Then there
exists a reduced envelope for H in G.
We will apply Reid’s result to show the following.
Proposition 5.6. Every open subgroup of Aut(∆) is commensurable with
the reduced envelope of a cyclic subgroup.
Proof. Let H be an open subgroup of Aut(∆) and assume without loss of
generality that J ⊆ S and H1 = H ∩ StabAut(∆) (RJ ) are as in Deﬁnition 4.16(iii). Let h1 be the hyperbolic element of H1 as in Deﬁnition 4.19,
so that Pc(h1 ) = WJ .
By Theorem 5.5, the group hh1 i has a reduced envelope E in Aut(∆).
In particular, [E : E ∩ H1 ] is ﬁnite.
On the other hand, H2 := E ∩ StabAut(∆) (RJ ) is an open subgroup of G
containing hh1 i, hence Lemma 4.27 applied on H2 shows that H2 is a ﬁnite
index subgroup of StabAut(∆) (RJ ) for the same subset J ⊆ S, i.e.,
[StabAut(∆) (RJ ) : StabAut(∆) (RJ ) ∩ E] < ∞.
Since also H1 has ﬁnite index in StabAut(∆) (RJ ) by Lemma 4.27 again, it
follows that also [H1 : H1 ∩ E] is ﬁnite. We conclude that H1 , and hence
also H, is commensurable with E, which is the reduced envelope of a cyclic
subgroup.

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