arXiv:1610.04143v1 [] 13 Oct 2016

Property Pnaive for acylindrically hyperbolic groups
Carolyn R. Abbott, François Dahmani
Abstract
We prove that every acylindrically hyperbolic group that has no
non-trivial finite normal subgroup satisfies a strong ping pong property,
the Pnaive property: for any finite collection of elements h1 , . . . , hk ,
there exists another element γ 6= 1 such that for all i, hhi , γi = hhi i ∗
hγi. We also obtain that if a collection of subgroups H1 , . . . , Hk is a
hyperbolically embedded collection, then there is γ 6= 1 such that for
all i, hHi , γi = Hi ∗ hγi.

The Ping-Pong argument is an iconic argument in group theory, which
works particularly well in hyperbolic settings. It allows one to produce free
subgroups at will, in many groups. However there are several variations of
it, and of this production of free groups. One of the strongest ping-pong
properties of a group is property Pnaive , defined in [1].
Pnaive stands for “naively permissive”: a group satisfying it “(...) is so
free that, for any finite subset F of G\{1} there exists a partner γ0 of infinite
order in G such that each pair {x, γ0 } with x ∈ F generates a subgroup which
is as free as possible.” ([1]). “As free as possible” means that the generated
subgroup is canonically isomorphic to hxiG ∗ hγ0 iG ' hxiG ∗ Z.
Property Pnaive is known for Zariski-dense discrete subgroups of connected simple groups of rank 1 with trivial center [1, Thm 3], hyperbolic
groups and relatively hyperbolic groups without non-trivial finite normal
subgroup, and non-Euclidean cubical CAT(0) groups without non-trivial
finite normal subgroup [6].
The class of acylindrical hyperbolic groups, whose systematic study was
initiated by Osin [7] after studies in [4, 5, 2], has attracted some interest due
to its many examples.
The action of a group G on a metric space X is acylindrical if for all  > 0
there exist constants M, N ≥ 0 such that for all x, y ∈ X with d(x, y) ≥ M ,
the number of elements g ∈ G satisfying d(x, gx) ≤  and d(y, gy) ≤  is
at most N . One says that a group is acylindrically hyperbolic if it admits a
non-elementary acylindrical, cobounded action by isometries on a hyperbolic
space.
1

M. Sageev asked one of us whether the property Pnaive is general among
all acylindrically hyperbolic groups, provided that they don’t contain a nontrivial finite normal subgroup (an obvious obstruction). This includes mapping class groups, Out(Fn ), some models of infinitely presented random
groups, and many other examples.
Property Pnaive is known to imply the simplicity of the reduced C ∗ –
algebra of the group. For acylindrically hyperbolic groups, this simplicity is
known by [4, Thm 2.35], but it was not established through the perspective of
the Pnaive property. However, the question of whether or not acylindrically
hyperbolic groups satisfy the Pnaive property was implicit in [4], and thus a
legitimate clarification is desirable.
In this note, we report that all acylindrically hyperbolic groups without
non-trivial finite normal subgroups indeed have property Pnaive (Theorem
2.3).
Recall that given a group G acting by isometries on a hyperbolic space
X, an element g ∈ G is called elliptic if the orbit g · s is bounded for some
(equivalently any) s ∈ X, and it is called loxodromic if the map Z → X
defined by n 7→ g n s is a quasi-isometric embedding for some (equivalently
any) s ∈ X. A subgroup is called elliptic if all its elements are elliptic.
We actually prove a stronger statement (Theorem 2.2), of which the
Pnaive property is a consequence: given a group G without non-trivial finite
normal subgroups that acts acylindrically by isometries on a hyperbolic
space X and a collection of subgroups H1 , . . . , Hn which are elliptic in X,
there exists an element γ ∈ G such that hγ, Hi i ' hγi ∗ Hi .
To prove this statement, we find a loxodromic element in G whose axis
escapes all quasi-fixed-point sets for these subgroups. We then show that
a sufficiently large power of this loxodromic element plays ping-pong with
each of the given subgroups. Thus the proof of the Pnaive property for
acylindrically hyperbolic groups reduces to finding a hyperbolic space with
an acylindrical action of G in which all of the given elements are elliptic.

Acknowledgments
We thank M. Sageev for asking the question and encouraging us to write
this note. C.A. is supported by NSF RTG award DMS-1502553. F.D. is
supported by the Institut Universitaire de France. The authors would like
to thank the Mathematical Sciences Research Institute for hosting them
during the completion of this project.

2

1

Axes of loxodromic elements

In this section we establish useful facts regarding the axes of loxodromic
elements in acylindrical actions on hyperbolic spaces.
If γ is a loxodromic isometry on a geodesic δ–hyperbolic space X, its
axis A(γ) is the union of all bi-infinite δ–quasigeodesics converging to its
two fixed points in ∂X. It is convenient to call the two fixed points γ + and
γ − , where γ + is the attracting fixed point of γ.
Let G be a acylindrically hyperbolic group and consider an acylindrical
action of G on a δ–hyperbolic geodesic space X. For a subgroup H of G,
we denote by FixK (H) the set {x ∈ X | d(x, hx) ≤ K for all h ∈ H}.
Lemma 1.1. Let γ ∈ G be loxodromic on a δ–hyperbolic geodesic metric
space (X, d), and let A(γ) be its axis. If γ − ∈ Fix50δ (H) for some elliptic
subgroup H ⊂ G, then γ + ∈ Fix50δ (H) as well. Similarly, if γ + ∈ Fix50δ (H)
then γ − ∈ Fix50δ (H).
Proof. By acylindricity, if H is infinite, then Fix50δ (H) is bounded. Thus if
γ − ∈ Fix50δ (H), it must be that |H| < ∞. As Fix50δ (H) is K–quasi-convex
for some K, there is a quasi-ray R asymptotic to A(γ) with R− = γ − such
that R ⊂ Fix50δ (H). Thus there is a subray T of A(γ) with T − = γ − and a
constant D = D(K, δ) such that for each h ∈ H, d(hx, x) ≤ D for all x ∈ T .
As γ −N x ∈ T for N ≥ 0, it follows that for each h ∈ H,
d(x, γ N hγ −N x) = d(γ −N x, hγ −N x) ≤ D.
As T is an infinite ray, we can choose a point y ∈ T that is arbitrarily far
from x, and for all N , γ N hγ −N moves both x and y by at most D. However,
by the acylindricity of the action the number of such elements is uniformly
bounded, and so for each h ∈ H there is some N0 such that γ N0 hγ −N0 = h.
Thus, h ∈ CG (γ N0 ) and so dHaus (A(γ), hA(γ)) < ∞. Therefore, hγ + = γ +
for each h ∈ H.
We will show that γ + ∈ Fix50δ (h) for each h ∈ H. It will then follow
that γ + ∈ Fix50δ (H). Suppose γ + 6∈ Fix50δ (h) for some h ∈ H, and consider
a sequence (xi )i∈N converging to γ + . It follows that (hxi )i∈N converges to
hγ + . By assumption, there exists some K > 0 such that for all i ≥ K,
xi 6∈ Fix50δ (h), so for all i ≥ K, d(xi , hxi ) > 50δ. Therefore the Gromov
product (γ + · hxn )x0 ≤ 100δ, and so γ + 6= hγ + , a contradiction.
Given an isometry φ of a space X, we say that a point is r–quasi-minimal
for φ if d(x, φ(x)) ≤ inf y∈X d(y, φ(y)) + r.
3

Lemma 1.2. (Location of the axis) Let X be a geodesic δ–hyperbolic space,
and γ1 , γ2 two loxodromic isometries of X with disjoint pairs of fixed points
in ∂X. Let U be a neighborhood of γ1− and V be a neighborhood of γ2+ .
Then, for sufficiently large n, γ1n γ2n is a loxodromic isometry whose repelling
fixed point in is U and whose attracting fixed point is in V .
Proof. Let Ai be a 10δ–quasiline invariant by γi that consists of (all) 10δ–
quasi-minimal points for γi Let x be a closest point projection of A2 on A1 ,
and let y be a closest point projection of x on A2 . Let σ0 be a sub-segment
of A2 of length 1000δ starting at y, let y0 be its final point, and let σ be
γ2−n σ0 (see Figure 1).
γ1n y0
γ1n x

A2

γ1n y

y0
x

σ0

A1

y

σ

γ2−n y

Figure 1: Location of axes
Consider the path obtained by concatenation of the quasigeodesic segments:
p = [γ2−n y, y], [y, x], [x, γ1n x], [γ1n x, γ1n y], [γ1n y, γ1n y0 ].
Notice first that this concatenation is a (1, 4 × 45δ)–quasi-geodesic. To
prove that, it is sufficient to observe the control on the Gromov product at
the four joint points: all (γ2−n y · x)y , (y, γ1n x)x , (x, γ1n y)γ1n x , (γ1n x, γ1n y0 )γ1n y
are below 20δ, by definitions of the points (sometimes after translating the
configuration by γ1−1 ). This yields the quasi-geodesic property.
Now, we claim that γ2−n y is 100δ–quasi-minimal for γ2n γ1n .

4

Observe first a general fact in hyperbolic spaces: if a point p is not
100δ–quasi-minimal for an isometry φ of a δ–hyperbolic space, then for p0
a point at distance 100δ from p on a segment [p, φp], one has d(p0 , φp0 ) ≤
d(p, φp) − 190δ (we leave this fact to the reader).
We’ll apply this observation to p = γ2−n y and φ = γ1n γ2n . Since p0 ∈ γ2−n σ0
(for the p0 of the observation), we have φp0 ∈ γ1n σ0 . The path p being a
quasigeodesic guarantees that |d(p0 , φp0 ) − d(p, φp)| ≤ 4 × 45δ = 180δ. By
the previous general observation this proves γ2−n y is 100δ–quasi-minimal for
γ2n γ1n .
This claim implies that γ1n γ2n is loxodromic, because we found a point
that is 100δ–quasi-minimal and that is displaced by more than 800δ. Moreover, its 120δ–quasi-axis (consisting of 120δ–quasi-minimal points) contains
γ2−n σ (and therefore its image γ1n σ).
In other words, if A denotes the quasi-axis A of γ1n γ2n , the projection
of A on A2 reaches γ2−n σ. Thus, if n is large enough it means that the
repelling point of γ1n γ2n is in U . Similarily, looking at the projection of A on
A1 reaching the projection of γ1n σ, for n large enough, the attracting point
of γ1n γ2n is in V .
Here is the main proposition of this section.
Proposition 1.3. Let G be a group acting by isometries, coboundedly and
acylindrically on a non-elementary geodesic hyperbolic space (X, dX ). If
H1 , . . . , Hk ∈ G is a family of subgroups that are elliptic on X such that,
for all i, Fix50δ (Hi ) 6= X, then there exist infinitely many independent loxodromic elements whose axes have bounded intersection with each Fix50δ (Hi ).
Proof. Throughout the proof, let F (H) = Fix50δ (H). If Hi is an infinite order elliptic subgroup, then F (Hi ) is a bounded set, and the conclusion holds
for every loxodromic element. Thus it suffices to assume that H1 , . . . , Hk
have finite order. In this case, we will show that we can find infinitely many
loxodromic elements γ such that γ ± 6∈ F (Hi ) for all i. It will then follow
that Aγ ∩ F (Hi ) is bounded for all i.
Observe that since Fix50δ (Hi ) 6= X for all i and X is non-elementary, we
have that F (Hi ) 6= ∂X.
We will proceed by induction. Choose any loxodromic element, g1 ∈ G.
If g1± 6∈ F (Hi ) for all i, we stop. Otherwise, without loss of generality,
we may assume g1± ∈ F (H1 ). Choose another loxodromic element g2 ∈ G
with one (hence both) fixed point at infinity not in F (H1 ). This can be
found, since F (H1 ) is not the entire boundary, and the action of G on X
5

is cobounded. Assume without loss of generality that g2± ∈ F (H2 ). Let
U ⊂ ∂X be an set containing g2+ such that U ∩ F (H1 ) = ∅. By Lemma
1.2, there exists N0 such that for all n0 > N0 , (g1n0 g2n0 )+ ∈ U . Letting
γ0 = g1n0 g2n0 , it follows that γ0+ 6∈ F (h1 ). By Lemma 1.1, it follows that,
additionally, γ0− 6∈ F (H1 ). By a similar argument, increasing N0 if necessary,
γ0± 6∈ F (H2 ). If γ0± 6∈ F (Hi ) for all i, we stop.
Otherwise, without loss of generality, assume γ0± ∈ F (H3 ). As F (H3 ) is
not the entire boundary and the action of G on X is cobounded, we may
choose a a loxodromic element g3 with g3± 6∈ F (H3 ). If g3± 6∈ F (Hi ) for all i,
we stop. Else, we may assume g3± ∈ F (Hi ) for some
1.2, if

 i 6= 3. By Lemma
U1 ⊂ ∂X is an open set containing γ + with U1 ∩ F (H1 ) ∪ F (h2 ) = ∅, there
exists N1 such that for all n1 > N1 , (γ0n1 g3n1 )+ ∈ U1 . Letting γ1 = γ0n1 g3n1 ,
and using a similar argument for γ1− , by Lemma 1.1,
γ1± 6∈ F (H1 ) ∪ F (H2 )

and

γ1± 6∈ F (H3 ).

If γ1± 6∈ F (Hi ) for all i, we stop.
Otherwise, we continue this procedure, and after at most k − 2 iterations
we will have produced an element γ = γj for some 0 ≤ j ≤ k − 2 such
that γ ± 6∈ F (Hi ) for all i. To find infinitely many such elements γ, we need
nj
nj
only change the final step of the procedure. By construction, γ = γj−1
gj+2
where nj > Nj , where Nj is provided by Lemma 1.2. However, it is clear
from the construction that the exponents of γj−1 and gj+2 may be chosen
independently, so for each nj , n0j > Nj ,
n

n0

j
j
(γj−1
gj+2
)± 6∈ F (Hi )

for all i.

If at any step we chose a loxodromic element g = gj for some 0 ≤ j < k−2
such that g ± 6∈ F (Hi ) for all i, then we proceed as follows. Choose any h ∈
−1
G, and consider h−1 gh, a loxodromic element such that
= ∅.
S A(h gh)∩A(g)

k
+
Let U be an open set containing g such that U ∩
i=1 F (Hi ) = ∅. Then
by two applications of Lemma 1.1, there exists an N ≥ 0 such that for
each n ≥ N , (g −n h−1 ghg n )± ∈ U , and we have constructed infinitely many
loxodromic elements γ = g −n h−1 ghg n with γ ± 6∈ F (Hi ) for all i, as desired.

2

Ping-pong with everyone

Proposition 2.1. Let G be a group acting by isometries, and acylindrically
on a geodesic hyperbolic space (X, dX ).
6

Let G0 be a subgroup of G that is elliptic on X. Assume that γ ∈ G is
loxodromic on X and that the intersection of Fix50δ (G0 ) and the quasi-axis
A of γ has diameter at most D. If N is such that γ N translates by at least
10D + 1000δ, then the group generated by γ N and G0 is the free product of
hγ N i and G0 .
Proof. Let D0 be an upper bound on the diameter of the projection of
an axis of a conjugate of γ to the axis of another conjugate. Let ∆ =
max{D, D0 , 1000δ, K1 , K22 }, where K1 , K2 are constants so that in geodesic
δ–hyperbolic spaces, all K1 –local (1, 10δ)–quasigeodesics are K2 –global quasigeodesics.
Let us choose N such that the translation length of γ N is at least 10 ×
∆ + 1000δ.
Consider the (abstract) HNN extension H = G0 ∗{1} and its Bass-Serre
tree T . We will show that the natural homomorphism from H to hγ N , G0 i
sending G0 to G0 and a stable letter to γ N is injective. It then follows that
it is an isomorphism.
In order to show the injectivity, consider a normal form for an element
of H:
(1)
(2)
(k)
h = g0 tm1 g0 tm2 . . . g0 tmk
(i)

with mj ∈ Z\{0} except for j = k, and g0 ∈ G0 \{1} except for i = 1 (where
(1)
g0 can be 1 and mk can be 0). We are only interested in the conjugacy
(1)
class of h, and thus we may assume g0 6= 1.
Pick a base point x0 in Fix10δ (G0 ) and write
(1)

(j)

x2j−1 = g0 tm1 . . . g0 x0 ,
(1)

(j)

x2j = g0 tm1 . . . g0 tmj x0 .
(1)

(j)

Let us also write h2j−1 = g0 tm1 . . . g0 .
Consider geodesics pi = [xi , xi+1 ]. Thus for each even i, pi is a geodesic
of length at most 10δ, and for each odd i, pi is a geodesic containing a central
subsegment of length at least 10∆ + 950δ that is in the 10δ–quasi-axis of
−1
γ hi .
When i is odd, let us denote by Ri the distance from xi to the quasi-axis
−1
−1
of γ hi . Observe that Ri ≤ |pi |/2 − 5∆ − 400δ since the translation of γ hi
along its axis is at least 10∆ + 1000δ.
Observe also that |Ri −Ri+2 | ≤ 10δ, since the difference (hi γ m(i+1)/2 )−1 hi+2
between the conjugators is an element of G0 , and hence moves x0 by at most
10δ.
7

xi−1
Ri−2

pi−1

xi
Ri

xi+1
pi−2
pi
Ri−2
xi−2
Figure 2: If the Gromov product (xi+1 · xi−2 )xi is larger than Ri + 2∆ + 100δ
−1
−1
for i odd, then the quasiaxes of γ hi−2 (dotted, left) and of γ hi (dotted,
right) remain 10δ–close for a length at least 2∆.
Let us estimate the Gromov product (xi+1 · xi−2 )xi for i odd. Assume
it is larger than Ri + 2∆ + 100δ. Then it follows that pi−2 and pi remain
2δ–close to each other for a segment of length at least Ri + 2∆ + 50δ (see
Figure 2).
In particular, by the definition of Ri , we obtain that the quasi-axis of
−1
h−1
i−2
γ
and of γ hi remain 10δ–close for a length of at least 2∆.
By the definition of ∆, this implies that their axes are the same, and
((i−1)/2)
therefore that g0
preserves the axis of γ. Since it moves one point on
the axis a distance of at most 10δ, it moves all points on the axis by a distance
of at most 50δ, and this contradicts our assumption that Fix50δ (G0 ) ∩ A has
diameter at most D.
Thus we have established that (xi+1 · xi−2 )xi ≤ |pi |/2 − 2∆ + 100δ, and
similarily, (xi+1 · xi−2 )xi ≤ |pi−2 |/2 − 2∆ + 100δ.
Now take m2i+1 to be the midpoint of p2i+1 for all i and let m−1 =
1, m2k+1 = h. Then the concatenation of the geodesic segments [m2i−1 , m2i+1 ]
is a (2∆)–local (1, 10δ)–quasigeodesics, since the final segment of length
2∆ of [m2i−1 , m2i+1 ] and the initial segment of length 2∆ of [m2i−1 , m2i+3 ]

8

xi−1
pi−1

xi+1
pi+1

xi

x1+2

pi
mi

mi+2
mi−2
pi+2
pi−2

xi+3

xi−2
Figure 3: The concatenation of segments [mi , mi+2 ], for odd i, is a quasigeodesic.
are in p2i+1 . By choice of ∆, this concatenation is therefore a global K2 –
quasigeodesic. (See Figure 3.) Since its length is more than 4∆, its endpoints
are at a distance greater than 4∆/K2 − K2 , which is positive.
Theorem 2.2. Let G be a group with no non-trivial finite normal subgroup.
Let (X, d) be a geodesic, hyperbolic space upon which G act by isometries,
acylindrically, and coboundedly.
Then for every finite collection {H1 , . . . , Hk } of subgroups of G that are
elliptic in X, there exists an element γ ∈ G such that for all k, hγ, Hk i =
hγi ∗ Hk .
Proof. First we prove that for all elliptic subgroup H, Fix50δ (H) 6= X.
Assume that there is an elliptic subgroup H on X, for which Fix50δ (H) = X.
Then by acylindricity and the non-elementarity of X, this H must be finite.
But any conjugate H 0 has also the property that Fix50δ (H 0 ) = X, and
acylindricity forces the number of conjugates to be finite. It follows that the
normaliser NG (H) of H has finite index in G. There exists a finite-index
normal subgroup G0 of G contained in NG (H) (for instance the intersection
9

of the conjugates of NG (H)). Then H and all its conjugates by G are normal
in G0 , since they are images of H by automorphisms of G0 . Since there are
finitely many such conjugates, their product is still a finite normal subgroup
of G0 , and it is also normal in G. Hence G has a non-trivial finite normal
subgroup, which is contrary to our assumptions.
Therefore, for all elliptic subgroups H, Fix50δ (H) 6= X. One can thus
use Proposition 1.3: there is a loxodromic element γ and a constant D ≥ 0
such that A(γ) ∩ Fix50δ (Hi ) has diameter at most D.
Finally, by Proposition 2.1: for each i there is an Ni such that hγ Ni , Hi i '
N
hγ i i ∗ Hi for all i. Setting N = max{N1 , . . . , Nk }, the result follows.
The previous theorem applies to a wide variety of situations, for instance
a collection of quasi-convex subgroups of a hyperbolic group, a collection
of relatively quasi-convex subgroups of a relatively hyperbolic group, or a
hyperbolically embedded collection of subgroups of an acylindrically hyperbolic group, with respect to some generating set (in the sense of [4, Definition
4.25]).
We illustrate two particularly nice instances of these situations. The first
is property Pnaive .
Theorem 2.3. Let G be an acylindrically hyperbolic group with no nontrivial finite normal subgroup. Then G satisfies property Pnaive .
Proof. We first need the following lemma.
Lemma 2.4. Let G be a group acting by isometries and acylindrically on a
geodesic hyperbolic space (X, dX ).
Let g0 be an element of G. Then there exists a geodesic hyperbolic space
(Y, dY ), endowed with an isometric G–action that is cobounded and acylindrical, and in which g0 is elliptic.
If G1 , . . . , Gk are subgroups of G that are quasi-convex in G with respect
to the metric dX , and whose set of cosets is uniformly geometrically separated
for dX , then there exists a geodesic hyperbolic space (Y, dY ), endowed with
an isometric G–action that is cobounded and acylindrical, and in which each
Gi is elliptic.
Proof. Let us prove the first part of the statement. By acylindricity, g0 is
either elliptic or loxodromic (see for instance [3, Lemma 2.2]). If g0 is elliptic
on X, there is nothing to prove. Hence we assume it is loxodromic. Then
the cone-off construction over its quasi-axis, performed in [4, §5.3], provides
a suitable space Y by [4, Theorem 5.39, Prop. 5.40].

10

For the second part, we directly apply [4, Theorem 5.39, Prop. 5.40] to
get the result.
Let h1 , . . . , hk ∈ G be any elements of G. As G is acylindrically hyperbolic, it acts by isometries, coboundedly, and acylindrically on a geodesic
hyperbolic space (X, dX ). By repeatedly applying Lemma 2.4, we may assume that there is a geodesic metric space (Y, dY ) with an isometric G–action
that is acylindrical in which h1 , . . . , hk are all elliptic. We can then apply
Theorem 2.2 in order to get the conclusion.
The second instance deals with a collection of hyperbolically embedded
subgroups.
Theorem 2.5. Let G be a group with no non-trivial finite normal subgroup.
Let (X, d) be a geodesic, hyperbolic space upon which G act by isometries,
acylindrically, and coboundedly.
Then for every finite collection {H1 , . . . , Hk } of subgroups of G that is
hyperbolically embedded as a collection of subgroups, with respect to (X, d),
there exists an element γ ∈ G such that for all k, hγ, Hk i = hγi ∗ Hk .
Proof. In the metric (X, d), these groups are quasi-convex (see [4, Thm
4.31]) and uniformly geometrically separated [4, Thm 4.24 (a)]. We may
thus apply the Lemma 2.4, and Theorem 2.2 then establishes the result.
It is not clear when one can replace the assumption that the collection of
groups is hyperbolically embedded as a collection by the weaker assumption
that each group is hyperbolically embedded. If one has a collection of subgroups of an acylindrically hyperbolic group G, each of which is elliptic in a
non-elementary acylindrical action of G, when is there an acylindrical action
in which all are elliptic simultaneously? When can one find an element that
plays ping-pong with each of these groups?

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