CAT(0) Groups and Acylindrical Hyperbolicity

arXiv:1610.08005v2 [] 9 Jan 2017

Burns Healy
Abstract
In this paper we take a result stating that rank one elements of a CAT(0) group are
generalized loxodromics and expand it to show the reverse implication. This gives us, in
particular, a complete classification of the intersection of CAT(0) and acylindrically hyperbolic groups, and demonstrates exactly which elements are generalized loxodromics.
We go on to apply this classification to the braid groups and OutpWn q in order to learn
about their potential CAT(0) structures, ruling out the cases of Euclidean buildings
and symmetric spaces.

1

Introduction

As spaces, CAT(0) metric spaces have non-positive curvature, while hyperbolicity represents the property of strictly negative curvature. We will restrict our attention to
the groups that act geometrically on them. Generalizations of hyperbolicity, such as
relative hyperbolicity and acylindrical hyperbolicity, relax the strictness of this action.
Such a description leads us to the natural question: in what sense can these generalized hyperbolic metric actions also meet our criteria of non-positive curvature? A
more direct question: where does the class of CAT(0) groups intersect those which
are acylindrically hyperbolic? This question has generated a lot of interest, leading to
recent results in the category of CAT(0) cube groups and spaces in [CM16] and [Gen16].
This question is already answered in part in [Sis] and [Sis16], where Sisto gives
results which tell us that rank one isometries of CAT(0) groups are generalized loxodromics, which implies that they are acylindrically hyperbolic. We go one step further
and make the claim that these classes of elements completely coincide.
Theorem 2.7 Let G be a group, which is not virtually cyclic, acting geometrically
on a CAT(0) space X. Then G is acylindrically hyperbolic if and only if it contains an
element g which acts as a rank one isometry on X.
This classification will be useful to us in order to help classify types of groups
suspected to be CAT(0). We begin by looking at the braid groups. Charney originally
posed the question are all finite type Artin groups CAT(0)? This is still an open
question for the braid groups, although the answer is known for small index. For
n ď 3, it is easy to see by looking at the algebraic description of the group. The cases
of n “ 4 and 5 were proved by Brady and McCammond in [BM10], and the n “ 6
case was proved by Haettel, Kielak, and Schwer in [HKS16]. It was also proved by
Huang, Jankiewicz and Przytycki that the 4-strand braid group, though CAT(0), is
not CAT(0)-cubed [HJP16].

1

Our contribution here will be to combine Theorem 2.7 with a result of Bowditch
(see 3.1) to show:
Theorem 3.2 Let Bn be the braid group with n ě 4, and suppose that X is a
CAT(0) space on which Bn acts geometrically. Then Y is a rank one CAT(0) space
in the natural splitting of X “ Y ˆ R. In particular, Bn does not act on a Euclidean
building.
For more information about CAT(0) buildings, the reader is referred to [Dav98].
The fact that these two classes of CAT(0) spaces are disjoint is explored at length in
[BB08]. In particular, in all Euclidean buildings and symmetric spaces, any bi-infinite
geodesic bounds a half-flat.
There has also been some interest in whether the automorphism group of universal
right angled Coxeter groups admit a geometric action on a CAT(0) space, see [Cun15].
Using a similar method that shows the acylindrical hyperbolicity of OutpFn q, we have
the following.
Theorem 4.6 OutpWn q is acylindrically hypebolic for n ě 4.
Then, in the same way as braid groups, we obtain a relevant CAT(0) result as a
corollary.
Theorem 4.7 Suppose OutpWn q acts geometrically on X a CAT(0) space. Then
X contains a rank one geodesic. In particular, OutpWn q cannot act geometrically on
a Euclidean building.
The author would like to thank his Ph.D. advisor, Genevieve Walsh, for her continued support, as well as Kim Ruane and Adam Piggott for helpful suggestions and
comments.

2

Classification

We begin with some definitions.
Definition 2.1. A metric space action G œ S is called acylindrical if for every ǫ ą 0
there exist R, N ą 0 such that for any two points x, y P S such that dpx, yq ě R, the
set
tg P G|dpx, g.xq ď ǫ, dpy, g.yq ď ǫu
has cardinality less than N .
We quickly note here that this property does not imply properness. This still allows
individual points to have infinite stabilizers.
Definition 2.2. A group G is called acylindrically hyperbolic if it admits an acylindrical
action on a hyperbolic space X which is not elementary; that is, the group has a limit
set inside the boundary of the space X of cardinality strictly greater than 2.
It’s worth pointing out that this means we do not wish to consider groups which are
finite or virtually cyclic as being acylindrically hyperbolic, despite the fact that they
are (elementary) hyperbolic. These are the only hyperbolic groups we exclude, as any
non-elementary hyperbolic group will satisfy this requirement by the natural action on

2

its Cayley Graph.
Remark 2.3. [Osi16] For any acylindrical group action on a hyperbolic space, no elements act as parabolics. This means for such an action, the effect of any particular
group element is either loxodromic or elliptic.
Definition 2.4. Let G be an acylindrically hyperbolic group. An element g P G is
called a generalized loxodromic if there is an acylindrical action G œ S for S a hyperbolic
metric space such that g acts as a loxodromic.
The status of being a generalized loxodromic is a group theoretic property. While
one qualifying action might have a particular element acting loxodromically, in another
it may act elliptically. The existence of a generalized loxodromic can be taken as an
alternate definition of acylindrical hyperbolicity.
Osin asks under what conditions an acylindrically hyperbolic group might have a
uiniversal acylindrical action, which is one in which all generalized loxodromics act
as loxodromics. Abbott gives an example of a finitely generated (but not finitely presented) group with no such action in [Abb16].
Before getting to our main theorem, we will recall a few definitions.
Definition 2.5. [BH99] Let X be a geodesic metric space, and ∆ a geodesic triangle.
¯ be a comparison triangle in E2 , i.e. a geodesic triangle with the same side lengths.
Let ∆
¯
We say that X is CAT p0q if, for all x, y P ∆ and all comparison points x̄, ȳ P ∆
dX px, yq ď dE2 px̄, ȳq.
and this holds for all ∆.
Definition 2.6. [CS15] A geodesic γ in a CAT p0q space X is said to be contracting if
there exists a constant D ą 0 such that for all x, y P X
dX px, yq ă dX px, πγ pxqq ùñ dX pπγ pxq, πγ pyqq ă D
Equivalently, any metric ball B that doesn’t intersect γ projects to a segment of length
ă 2D on γ.
Theorem 2.7. Let G be a group, which is not virtually cyclic, acting geometrically on
a CAT(0) space X. Then G is acylindrically hyperbolic if and only if it contains an
element g which acts as a rank one isometry on X. Furthermore, the set of generalized
loxodromics is precisely the set of rank one elements.
Proof. (ð) This follows from theorem 5.4 in [BF09], where it is stated that a geodesic
in a CAT(0) space is contracting exactly when it fails to bound a half flat, meaning rank
one geodesics are contracting. Next, contracting elements satisfy a property labelled
weakly contracting, shown in [Sis]. Sisto goes on to prove in Theorem 1.6 that this
property is strong enough to show that this element is contained in a virtually cyclic
subgroup, labelled Epgq, which is hyperbolically embedded in the group. This is one
of four equivalent conditions for being a generalized loxodromic, listed in Theorem 1.4
of [Osi16].
(ñ) If G is acylindrically hyperbolic, then it contains at least one generalized
loxodromic. This is because we can take the action on a hyperbolic space guaranteed by

3

the definition of acylindrical hyperbolicity, and knowing that it is devoid of parabolic
elements, invoke the non-elementary condition on the action to verify at least one
element must act as a loxodromic. Call this element g. We know by a result of Sisto
that this element is Morse in G [Sis16]. An equivalence in the setting of CAT(0) groups,
proved in [CS15], says that a (quasi-)geodesic in a CAT(0) space is contracting if and
only if it is Morse and if and only if it is rank one. The geometric nature of our action,
which says our space is quasi-isometric to our group, guarantees that because our group
element is Morse, its axes are as well. Therefore our element g has axes which are rank
one, i.e. it acts as a rank one isometry.
This equivalence allows us to restate the Rank Rigidity Conjecture for CAT(0)
groups, originally posited by Ballman and Buyalo.
Conjecture 2.8. Rank Rigidity Conjecture [BB08]
Let X be a locally compact geodesically complete CAT(0) space and G a discrete group
acting geometrically on X. If X is irreducible, then either:
• X is a Euclidean building or higher rank symmetric space
or
• G is Acylindrically Hyperbolic.

3

Braid Groups

Braid groups, which we will denote here by Bn , are an important example of groups
that are intermediate between hyperbolic and flat. They are not hyperbolic (nor even
relatively hyperbolic); indeed they have a number of flats. However, they do have
free subgroups and many properties shared by groups which are non-positively curved.
The following is obtained by Bowditch in [Bow08] by noting that Bn :“ Bn /ZpBn q
represents the mapping class group of a punctured surface.
Theorem 3.1. [Bow08] Let n ě 4. The group Bn :“ Bn /ZpBn q is acylindrically
hyperbolic.
A stronger statement holds. A result from [CW16] shows that all Artin-Tits groups
of spherical type, otherwise known as generalized braid groups, are acylindrically hyperbolic, after modding by their center.
Now this can be combined with 2.7 to show:
Theorem 3.2. Let Bn be the braid group with n ě 4, and suppose that X is a CAT(0)
space on which Bn acts geometrically. Then Y is a rank one CAT(0) space in the
natural splitting of X “ Y ˆ R. In particular, Bn does not act on a Euclidean building.
This holds for all Artin-Tits group of spherical type. By combining Theorem 2.7
to a result of Calvez and Wiest [CW16], we get that any CAT(0) space acted on geometrically by Artin-Tits group of spherical type must be rank one.
To illustrate this theorem, we give an example from an explicit CAT(0) complex. In
[Bra00], Brady constructs a 2-complex that B4 {ZpB4 q acts on made up of equilateral

4

triangles which is CAT(0) using the standard piecewise Euclidean metric. This is
obtained by ‘projecting’ down the infinite cylic factor corresponding to
ZpB4 q – Z “ă σ1 σ2 σ3 ą
The link of any vertex in this projected complex looks as in Figure 1.

a`

b´

c`

c´

d`

a´

f`

d´

e`

e´

c`

b`

f´

a`

Figure 1: An Arbitrary Link
Importantly, we note that the top right and bottom left vertices of this link are
identified, as well as the top left and bottom right. We recognize this as a 1-skeleton of
a CW-complex homeomorphic to a Möbius strip. Because this complex is CAT(0), this
link has a standard CAT(1) metric on it [BH99]. Because the corresponding triangles
are equilateral, this metric assigns each edge in this link a length of π3 . We now examine
the vertices labelled d´ and b´ , specifically noting that
dlk pb´ , d´ q “

4π
ąπ
3

Because this angle is larger than π, that means that the path obtained in the space
obtained by concatenating the paths from x0 to those points in the link is a local
geodesic. We note this path is also a portion of an axis for the group element bd´1 .
(Note: one of these elements must have an inverse, as both vertices have a plus, which
means that group element takes us towards x0 .)
This space enjoys the property that all vertices have isometric links, so if we look
at the link of the vertex b.x0 , we see the path the axis of bd´1 takes is in through b`
and out through d´ , which also have distance greater than π. This implies our axis,
which we will label γ :“ γbd´1 , is also a local geodesic at this vertex. Combining this
with the fact that edges of triangles in our space are local geodesics because the metric
is CAT(0) and locally Euclidean, this tells us that γ is a geodesic axis for the action
of the group element bd´1 .
Indeed this axis is rank one, as [BB08] proves the Rank Rigidity Conjecture in the
case where the dimension of the complex is 2. Because X is two dimensional, if γ
bounded a space isometric to E2` , then any vertices of links it went through would
have to have diameter at most π, as the link of that vertex would contain faces forming
a half-disk portion of the copy of E2` . As this axis is rank one, we know that our group
element is a generalized loxodromic, by Theorem 3.2.

5

So what does it look like? If we translate bd´1 into our standard generating set,
noting that σ1 , σ3 commute, we get the element
σ2 σ1 σ3 σ2´1 σ1´1 σ3´1

Figure 2: The element bd´1

4

OutpWnq

Assume here that n ě 4.
We will use the conventions in this section that
Fn “ xxi | y, Wn “ xwi | wi2 y
As wi´1 “ wi we will suppress inverse notation when working in Wn
We will make use of the following result, so it is helpful to list it here.
Theorem 4.1. [GPR12]
AutpWn q “ Aut0 pWn q ¸ Σn “ pWn ¸ Out0 pWn qq ¸ Σn
where the Wn factor is the whole of InnpWn q, Aut0 is partial conjugations, Out0 is
the image of Aut0 after quotienting out the inner automorphisms, and Σn is the full
symmetric group on n letters, corresponding to permuting the generators.
Lemma 4.2. Consider the subgroup Wn ą G :“ xw1 wi |i P t2, . . . nuy. Then G – Fn´1 .
Proof. It is clear that none of the n ´ 1 generators are redundant. Therefore, we are
reduced to demonstrating there are no relations. We begin by noting that
pw1 wi q´1 “ wi w1

6

Next, note that cancellation can only happen in the form wi wi “ 1, as these are
the only relators in Wn . This is only the case if we have wi w1 w1 wj , in which case this
is equal to wi wj , which is irreducible and simply another expression for the element
xi´1 xj´1 , or w1 wj wj w1 , which translates to xj´1 x´1
j´1 , i.e. a free reduction in Fn´1 .
This gives a natural map ι : AutpWn q Ñ AutpFn´1 q because an automorphism of
Wn will induce a map on the elements w1 wi , noting that this subgroup is characteristic. The subgroup G above is sometimes referred to as the ‘even subgroup’. Because
all cancellation happens in pairs, it is well defined to speak of words of even length
(including the empty word).
Using this characterization, it is easy to see that this subgroup must be characteristic. Given our decomposition of AutpWn q, we see that generators come in the form
of graph automorphisms and partial conjugations. In the former case, all wi are sent
to words of length one, and in the latter, wi are sent to either words of length one
or three. In either case, after possible cancellation in pairs, words of even length will
remain of even length. Because this subgroup is precisely the words of even length, a
property preserved under automorphisms of Wn , it is a characteristic subgroup.
Lemma 4.3. This map is injective.
Proof. Let φ : Wn Ñ Wn be an automorphism that fixes pointwise the set tw1 wi u.
Our goal is then to show that it must be the case that
φ “ idAutpWn q .
We do this by considering φ on each generator. If φ fixes each generator, it is
necessarily the identity.
• Suppose φpw1 q “ z ‰ w1 . Assume z is fully reduced. We know that z ‰ id,
as this is an automorphism. Because φ fixes w1 wi , it must be the case that for
all i, φpwi q “ z ´1 w1 wi . We invoke the fact that AutpWn q is generated only
by conjugations and permutations, following from the decomposition above, to
observe that the word we map to must begin and end with the same letter after
being reduced. If z ´1 does not end with the letter w1 (meaning z starts with
the letter w1 ), then z ´1 must start with the letter wi for each i. This is a
contradiction.
Therefore we assume z starts with the letter w1 . The last letter of z will be the
first letter of z ´1 w1 wi , which again because AutpWn q has no transvections, must
be either wi or empty. Because the former is impossible for all i simultaneously,
it must be that it is empty. This tells us that z “ w1 , which contradicts our
assumption.
• Now suppose φpwi q “ zi ‰ wi for some i ‰ 1. Then because φpw1 wi q “ w1 wi ,
we get that φpw1 q “ w1 zi´1 . This must be true for every i, meaning that for all
i ‰ 1, φpwi q “ zi “ zj “ φpwj q. This contradicts our assumption that φ is an
automorphism, because the image of the generators is no longer a generating set.

Due to the injectivity of this map, we will label it
ι : AutpWn q ãÑ AutpFn´1 q

7

Because our goal is to say something about OutpWn q, we look at what happens to
elements of InnpWn q. While it is not quite true that they map into OutpFn´1 q, we find
this is almost the case.
Lemma 4.4. Let r P AutpFn q be defined by rpxi q “ x´1
i . Then
ιpInnpWn qq Ă InnpFn´1 q ¸ tru.
Proof. Recall xj :“ impw1 wj`1 q. By abuse of notation, and because the map is injective, we will switch between these words in our computations. Denote C xi for conjugation by xi in AutpFn´1 q.
We will consider the effect that an inner automorphism of Wn will have on tw1 wi u,
recalling that this set of automorphisms is generated by elementary conjugations. These
will come in two flavors, denoting an inner conjugation by C j for conjugation of all
generating elements by wj :
• C 1 pw1 wj q “ wj w1 “ pw1 wj q´1 . Therefore C 1 “ r.
• C i, i ‰ 1
C i pw1 wj q “ wi pw1 wj qwi
“ wi w1 pw1 w1 wj w1 qw1 wi
“ wi w1 pwj w1 qw1 wi
“ wi w1 pw1 wj q´1 w1 wi
“ pw1 wi q´1 pw1 wj q´1 pwi w1 q´1
“ r ˝ C x1 pxj q

Finally, this gives us the following fact:
ῑpOutpWn qq Ă AutpFn´1 q{pInnpFn´1 q ¸ truq Ă OutpFn´1 q{xxRyy
where R “ xr|r 2 y – Z{2Z.
This relationship is summarized in the following diagram, where qr represents the
quotient map killing the normal closure of R:

AutpWn q

ι

AutpFn´1 q
q

q̄

OutpWn q

OutpFn´1 q
ῑ

qr

OutpFn´1 q{xxRyy
Figure 3: Diagramatic Relations of the Groups

8

Lemma 4.5. For any φ P AutpWn q,
rr, ιpφqs Ă InnpFn´1 q
Proof. We break this into cases, depending on what kind of automorphism φ is. We
need only consider the case where φ is a generator, because the inner automorphisms
are normal. To demonstrate this for a normal subgroup K Ÿ G, and an element r P G,
assume that a, b P G are such that rr, as P K, rr, bs P K. We’d like to show rr, abs P K.
Then,
rr, abs P K
õ
rabrb´1 a´1 P K
õ
brb´1 r “ k1 P K
1
´1
rark a P K
õ
conjugate by a´1 ; K is normal
´1
1
a rark P K
õ
´1
a rar P K
õ
conjugate by a
rara´1 P K
and we note the last line is true by hypothesis. Now, let’s look at the cases:
• φ is a graph automorphism (i.e. it permutes the generators), so φ P Σn . This
subgroup is generated by transpositions of wi . We further break this into subcases:
– φ “ pwi wj q, i ‰ 1 ‰ j. Then it is easy to see ιpφq permutes xi´1 , xj´1 and
that this map commutes with inverting every generator.
– φ “ pw1 wi q. In this case, φpw1 wj q “ wi wj “ wi w1 w1 wj for i ‰ j and
´1
φpw1 wi q “ wi w1 . Then ιpφqpxj q “ x´1
i´1 xj for j ‰ i´1 and ιpφqpxi´1 q “ xi´1 .
The rest of the proof that ιpφq commutes with r is left to the reader.
• φ is a partial conjugation. Once more, we are relegated to subcases:
– Neither the acting letter nor the acted-on letter is w1 . Then for i ‰ 1 ‰
j, φpwj q “ wi wj wi and φ fixes all other generators. Then ιpφq fixes all
generators of Fn´1 except xj´1 , and ιpφqpxj´1 q “ xi´1 x´1
j´1 xi´1 . Note that φ
is order two in the codomain, so ιpφq is also order two. This is borne out by
performing the calculation on the right hand side. We see that rr, ιpφqspxi q “
xi for i ‰ j ´ 1.
ιpφq ˝r ˝ ιpφq ˝rpxj´1 q “ ιpφq ˝r ˝ ιpφqpx´1
j´1 q
´1
“ ιpφq ˝rpx´1
i´1 xj´1 xi´1 q

“ ιpφqpxi´1 x´1
j´1 xi´1 q
“ xj´1
– The acting letter is w1 . Then call the acted on letter wi , so that φpwi q “
w1 wi w1 and fixes other generators. In this case, ιpφq inverts xi´1 and fixes
the other free generators. This automorphism commutes with inverting all
generators.

9

– The acted on letter is w1 , so φpw1 q “ wi w1 wi . Quick calculations show that
ιpφqpxj q “ x´2
i´1 xj . Then
ιpφq ˝r ˝ ιpφq ˝rpxj q “ ιpφq ˝r ˝ ιpφqpx´1
j q
2
“ ιpφq ˝rpx´1
j xi´1 q

“ ιpφqpxj x´2
i´1 q
2
“ x´2
i´1 xj xi´1
x´2

“ xj i´1 .

This allows us to make the observation that
R Ÿ ι ˝ qpAutpWn qq ă OutpFn´1 q.
In other words
xxRyy X ι ˝ qpAutpWn qq “ R.
More to the point, this allows us to replace Figure 3 with Figure 4.

AutpWn q

ι

impιq
q

q̄

OutpWn q

impι ˝ qq
ῑ

AutpFn´1 q
q

OutpFn´1 q

qr

impι ˝ qq{R
Figure 4: Involution Normality in the Image
It is shown in [BBF15] that OutpFn q is acylindrically hyperbolic. This is proven
by way of its action on the free factor complex, although it is unknown if this action
is itself acylindrical. Hyperbolicity of this complex is demonstrated in [BF14]. In
this action fully irreducible elements of OutpFn q act with Weak Proper Discontinuity
(WPD), which tells us that they are generalized loxodromics (in an action on a different
space). Theorem H in [BBF15] constructs a new action on a space quasi-isometric to
a tree, which we will denote by Q, that satisfies the conditions required by acylindrical
hyperbolicity, in which these same fully irreducible elements act loxodromically. The
fact that this action is acylindrical is stated in the discussion after Theorem I [BBF15].
Furthermore we are guaranteed, again by [BBF15] that all fully irreducible group
elements act loxodromically in this action on Q.
Theorem 4.6. OutpWn q is acylindrically hyperbolic, for n ě 4.
Proof. We will abuse notation throughout this proof, letting r represent both the
automorphism in AutpFn´1 q and its image under the map q.
The first thing we will do is make a slight modification to Q. Unlike in uniquely
geodesic spaces such as CAT(0) spaces, fixed point sets in arbitrary hyperbolic spaces

10

aren’t as nice as we like, so we will add in a little extra structure. Let δ represent the
constant of hyperbolicity for Q
Define Q̂ :“ Q Y E, where E consists of combinatorial edges of length δ between
any two points which are at distance at most δ in Q. We note that these two spaces are
quasi-isometric by noting that Q embeds into Q̂ in the natural way such that distances
are not changed, and the embedding is δ quasi-onto. The group OutpFn q will act on
Q̂ in the natural way on the embedded copy Q, and permute the edges in E according
to their endpoints.
Label Q̂R the fixed point set of R “ xry acting on Q̂.
Now define an action OutpWn q œ Q̂R . We start by noting there is a natural action
of M :“ ι ˝ qpAutpWn qq œ Q̂, because it is a subgroup of OutpFn´1 q. Now, to say
what an element of ψ P OutpWn q does, we look at its image, ῑpψq P qr pM q. Using
the structure of Figure 4, ῑpψq “ gR for some element g P OutpFn´1 q. Now, for any
f P Q̂R , we can define
ῑpψq.f “ g.f.
This is well-defined, because no matter what element of R we pick (i.e., either id or r),
they both have the same effect on f , (i.e., r.f “ f by definition of Q̂R .)
Finally we claim the image of OutpWn q leaves Q̂R invariant set-wise. That is to
say, doesn’t take it off itself. Let C P impι ˝ qq which encompasses any element coming
from OutpWn q, and let f P Q̂R . Note that from 4.5 that rr, Cs “ 1 in OutpFn´1 q. Then
r C .f “ C r.f
“ Cpr.f q
“ C .f.
So the point f is moved to under C is indeed fixed by r, as r C .f “ C .f .
Now in order to show that this action on Q̂R satisfies acylindrical hyperbolicity, we
must show three things:
1. Q̂R is hyperbolic.
2. This action satisfies acylindricity.
3. This action is non-elementary.
For the first task, we recall that hyperbolicity is a quasi-isometry invariant, so we
know that Q̂ is hyperbolic. We claim that Q̂R is quasi-convex in Q̂, making it also
hyperbolic. To show quasi-convexity, let f0 , f1 be in Q̂R such that a geodesic between
them leaves Q̂R . If such points don’t exist, our subspace is directly convex. Otherwise, label x0 , x1 the points (possibly the same as fi ) such that the chosen geodesic
first leaves then re-enters Q̂R . Let λ “ rx0 , x1 s, which by assumption only intersects
Q̂R in the endpoints. If we take r.λ, we obtain another, distinct geodesic between
x0 , x1 . Label this point xi . By the closeness of geodesics with the same endpoints in a
hyperbolic space, the distance between xi and g.xi is bounded by δ. Therefore there
is a combinatorial edge between them of length δ. Because r is order 2, it acts by
inversion on this edge, and therefore fixes its midpoint. This means that every point
on this geodesic is within distance at most 2δ of a fixed point. So this (and therefore

11

any) geodesic between points in Q̂R lies in a 2δ neighborhood of Q̂R . This means Q̂R
is 2δ quasiconvex, and therefore hyperbolic.
For acylindricity, we begin by letting Rpǫq, N pǫq be constants depending on ǫ that
demonstrate the acylindricity of the action OutpFn´1 q œ Q. We note that these same
constants will work to demonstrate acylindricity of impι ˝ qq because the relevant set of
elements will be a subset of the one we consider in the supergroup. Our claim is that
these same constants will once again work for impι˝qq{R. We proceed by contradiction.
Let ǫ ą 0, and Rpǫq as above. Then suppose
|tφ P impι ˝ qq{R|dpx, φ.xq ď ε, dpy, φ.yq ď εu| ě N.
Now consider the set of pre-images tqr´1 pφqu of these elements. Because qr is surjective, this set has no fewer elements than the original. Furthermore, because our
quotient is by R, which acts trivially on QR , these elements also have the same induced action. Therefore, this violates the assumption that there are fewer than N pǫq
elements in impι ˝ qq that satisfy this property. Finally, adding these combinatorial
edges to QR does not change the property of acylindrical hyperbolicity; it slightly
modifies the constants. This is because it does not change the distance of points in
Q, and elements moving the new combinatorial edges close to themselves must bring
those endpoints, which belong to Q, close to themselves. Specifically, for x, y P Q̂ with
distance dpx, yq ě R , the set
tg P G|dpx, g.xq ď ǫ, dpy, g.yq ď ǫu
is contained in the set, for x, y P Q with dpx, yq ě Rpǫ ` 2δq,
tg P G|dpx, g.xq ď ǫ ` 2δ, dpy, g.yq ď ǫ ` 2δu
which is finite by assumption.
Finally, we are tasked with showing this action is non elementary. In order to
demonstrate that the limit set is not elementary, we will show that there are two elements in the image of AutpWn q that act as loxodromics on Q̂. Due to quasi-convexity,
any element which acts as a loxodromic on Q and fixes Q̂R set-wise will also act as a
loxodromic on Q̂R , so establishing a loxodromic action on Q is sufficient. To find these
elements, we recall that Q is designed such that any elements acting loxodromically
and with WPD on the free factor complex also act as such on Q. To find these, we
turn to [BF14], which tells us that the elements which act loxodromically are exactly
those automorpshisms that don’t fix (or have any power which fixes) any individual
free factor. It is clear why this is a necessary condition, but sufficiency is more intricate.
We’ll fix some notation for partial conjugations in AutpWn q. Let
Pi,j pwi q “ wj wi wj
Pi,j pwk q “ wk

12

k‰i

Note that this element is order 2. To see what effect this has on the free group:
Pi,j pxi´1 q “ Pi,j pw1 wi q
“ w1 wj wi wj
“ w1 wj wi pw1 w1 qwj
“ pw1 wj qpwi w1 qpw1 wi q
“ xj´1 x´1
i´1 xj´1
Now let’s examine the product Pi,j ˝ Pi,k for k ‰ j.
Pi,k pPi,j pxi´1 qq “ Pi,k pxj´1 x´1
i´1 xj´1 q
“ xj´1 xk´1 xi´1 xk´1 xj´1 :
We further see that Pi,j commutes with Pk,n exactly when i ‰ k. For the following
element, consider all addition performed in the indices to be done mod n ´ 1. To find
a desired loxodromic element, let
µ “ Π1ďiďn´1 Pi,i`1 Pi,i`2 .
This is fully irreducible because it affects each generator, ranging on indices 1 ď i ď
n ´ 1, sending each one to a word as in :, which are not expressible as a conjugation.
We need one more loxodromic, which does not commute with µ, to finish the proof.
Let
τ “ Π1ďiďn´1 Pi,i`3 Pi,i`2 .
Defining this element, we see why at the beginning of the section we required that
n ě 4. Now that we have these two distinct loxodromics, we get a limit set strictly
greater than 2 (the endpoints of these elements), which completes the proof.
Theorem 4.7. Suppose OutpWn q acts geometrically on X a CAT(0) space. Then X
contains a rank one geodesic. In particular, OutpWn q cannot act geometrically on a
Euclidean building.
This follows from 2.7 and 4.6.

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