On the kernel of actions on asymptotic cones

2.1 Cayley graphs and 𝛿-hyperbolic spaces

In this subsection, we briefly recall the definitions of Cayley graphs and 𝛿-hyperbolic spaces. We also introduce our notation.

Then it is an easy fact that Γ ⁢ ( G , S ) is connected, and locally finite whenever 𝑆 is a finite set. If 𝑆 and 𝑇 are finite generating sets for a group 𝐺, then usually Γ ⁢ ( G , S ) is different from the graph Γ ⁢ ( G , T ) , but they are quasi-isometric when both 𝑆 and 𝑇 are finite.

Clearly, a group 𝐺 acts on the Cayley graph Γ ⁢ ( G , S ) by g ⋅ x = g ⁢ x . With the metric d S ⁢ ( g , h ) = ∥ g − 1 ⁢ h ∥ S , where ∥ x ∥ S means the metric between x ∈ G and the identity 𝑒 in the graph Γ ⁢ ( G , S ) , this action is isometric and cocompact. When we write the word metric, we sometimes use ∥ x ∥ when there is no confusion about the generating set, or when the generating set is not important.

Now we recall 𝛿-hyperbolic spaces and related notions.

A 𝛿-hyperbolic space 𝑋 has a typical and well-studied boundary called the Gromov boundary, denoted by ∂ X . To define the Gromov boundary, we first recall the Gromov product. For a triple x , y , z in 𝑋, the value

is called the Gromov product of three points x , y , z .

Now we define the Gromov boundary using the Gromov product. First, we say that a sequence [ x n ] in 𝑋 converges at infinity if

for fixed p ∈ X . Let 𝒮 be a set of sequences that converge at infinity. We declare [ x n ] ≈ [ y n ] if and only if

Then the relation ≈ is an equivalence relation. The Gromov boundary of 𝑋, ∂ X , is defined by ∂ X : = S / ≈ .

It is a fact that [ x n ] ≉ [ y n ] if and only if sup m , n ( x m | y n ) p < ∞ , and the Gromov boundary does not depend on the choice of the basepoint 𝑝.

Let us review the terminology of isometry on a 𝛿-hyperbolic space 𝑋. An isometry 𝑓 on 𝑋 is called elliptic if the orbit of 𝑓 is bounded for some, equivalently any, orbit. It is called parabolic if 𝑓 has exactly one fixed point in the boundary ∂ X . Lastly, we say that 𝑓 is loxodromic if it has exactly two fixed points in the boundary. Recall that 𝑓 is loxodromic if and only if the map Z → X given by n ↦ f n ⋅ x is a quasi-isometric embedding for any basepoint 𝑥 in 𝑋. We say that two loxodromic elements f , g are independent if their fixed point sets are disjoint.

Suppose that a group 𝐺 acts isometrically on a 𝛿-hyperbolic group 𝑋. Then there exists a canonical 𝐺-action on ∂ X . For g ∈ G and [ x n ] ∈ ∂ X , define

Then this action is well-defined.

2.2 Asymptotic cones

Asymptotic cones are the main ingredient of this paper, so we briefly recall their definition. Since asymptotic cones are a special kind of ultralimit of metric spaces, we start with the definition of ultrafilters, which is a useful concept to define ultralimits.

Recall that, from the first and second conditions, it cannot happen that both A ∈ ω and N − A ∈ ω . Now we will define the ultralimit of a sequence in a metric space. Using this, we can define an ultralimit of metric spaces.

Similarly to the usual limit, if an ultralimit exists, then it is unique, but in general, the ultralimit may not exist. However, when a metric space 𝑋 is compact, it is known that the ultralimit always exists. This implies that the ultralimit of any bounded sequence in ℝ always exists.

Suppose that a sequence { x n } converges to 𝑥 in the usual limit sense. Then, for any non-principal ultrafilter 𝜔, lim ω x n = x . So we can think of an ultralimit as a generalization of the usual limits, and this explains why we use non-principal ultrafilters. One of the difficulties of an ultralimit is that an ultralimit depends on the choice of an ultrafilter. Now we define ultralimits of metric spaces.

Clearly, the ultralimit of metric spaces depends not only on { ( X n , d n ) } but also the choice of the observation sequence { p n } and ultrafilter 𝜔.

Let 𝑋 be a 𝛿-hyperbolic space. Recall that both elements of the Gromov boundary of 𝑋 and of an asymptotic cone of 𝑋 can be considered as sequences in 𝑋. To avoid confusion, we denote an element of the Gromov boundary by [ x n ] ∈ ∂ X and an element of an asymptotic cone by { x n } ∈ Cone ω ⁡ ( X , d n , p n ) .

We define an asymptotic cone of 𝐺 by an asymptotic cone of its Cayley graph.

Since asymptotic cones are quasi-isometry invariant up to bi-Lipschitz, this definition is well-defined and we can omit the finite generating set 𝑆. Also, we can remove the observation sequence g n from the notation. Recall that a metric space 𝑋 is quasi-homogeneous if there exist a homogeneous space Y ⊂ X and a constant C > 0 such that, for any x ∈ X , d ⁢ ( x , Y ) < C . It is known that Cone ⁡ ( X , d n , p n ) does not depend on the choice of observation sequence when 𝑋 is quasi-homogeneous. Thus any asymptotic cone of a group does not depend on the choice of observation sequence { g n } , so we can simply write Cone ω ⁡ ( G , d n ) .

However, an asymptotic cone of a group still depends on the choice of a real sequence d n and ultrafilters. First, S. Thomas and B. Velickovic constructed a group with two distinct (non-homeomorphic) asymptotic cones [73]. Their group is only finitely generated, not finitely presented. Later, a finitely presented group with two non-homeomorphic asymptotic cones was constructed [56]. This group has a simply connected space (not a real tree) and a non-simply connected space as an asymptotic cone.

However, it is known that any asymptotic cone of hyperbolic groups is unique and it is a real tree. In general, the converse is not true (recall that a group is called lacunary hyperbolic if one of its asymptotic cones is a real tree), but when a group 𝐺 is finitely presented, then 𝐺 is hyperbolic if and only if one of its asymptotic cones is a real tree. Furthermore, A. Sisto completely classified asymptotic cones of hyperbolic groups as follows. If a real tree 𝑋 is the asymptotic cone of a group, then it is a point, a line, or the 2 ℵ 0 universal tree. We refer to [69, Corollary 5.9] for further details.

We end this subsection by introducing the natural action of 𝐺 on its asymptotic cone. Let 𝐺 be a group and consider one of its asymptotic cones Cone ω ⁡ ( G , d n ) . Note that an element in the asymptotic cone can be considered as an (admissible) sequence of 𝐺, so let { x n } ∈ Cone ω ⁡ ( G , d n ) . Define a group action by

It is straightforward that the group action is well-defined and isometric.

2.3 Acylindrically hyperbolic groups

In this subsection, we discuss acylindrically hyperbolic groups. Denis Osin first proposed this notion in [58]. It can be seen as a generalization of relatively hyperbolic groups. However, it still includes other significant groups in geometric group theory, e.g., the mapping class groups Mod ⁡ ( S ) , where 𝑆 is not an exceptional case and has no boundary cases, and the outer automorphism groups of the rank 𝑛 free group Out ⁡ ( F n ) . First of all, we introduce the definition of an acylindrically hyperbolic group.

Recall that an isometric action of 𝐺 on a 𝛿-hyperbolic space 𝑋 is called elementary if the limit set of 𝐺 on the Gromov boundary ∂ X contains at most 2 points.

Indeed, there are equivalent definitions for acylindrical hyperbolicity. The following theorem is well known and can be found in [58].

We note that 𝛿-hyperbolic spaces X 1 and X 2 in the aforementioned theorem may be different. The terminology WPD stands for “Weakly Properly Discontinuous” and was first introduced by Bestvina and Fujiwara [11]. We briefly recall its definition.

By the definition, if 𝑔 is WPD for some action 𝐺 on a 𝛿-hyperbolic space 𝑋, then 𝑔 must be a loxodromic element. Also, if 𝐺 acts acylindrically on 𝑋, then every loxodromic element is WPD. Recall that an acylindrically hyperbolic group contains infinitely many independent loxodromic elements [58]. For more details about the equivalence definitions, Theorem 2.10, we refer to [27]. For readers who are interested in WPD elements, we refer to [11, 64].

We say that a subgroup 𝐻 of 𝐺 is s-normal if the intersection g − 1 ⁢ H ⁢ g ∩ H is infinite for every g ∈ G . It is known that the class of acylindrically hyperbolic groups is closed under taking s-normal subgroups [58, Lemma 7.2]. By the definition of s-normal subgroups, every infinite normal subgroup is s-normal, so we obtain the following lemma.

Note that the space does not change in Lemma 2.13. In other words, if G ↷ X is non-elementary and acylindrical, then for any infinite normal subgroup H < G , the induced action H ↷ X is also non-elementary and acylindrical.

3.1 The unique maximal finite normal subgroup K ⁢ ( G ) and the amenable radical A ⁢ ( G )

First, we will discuss the unique maximal finite normal subgroup K ⁢ ( G ) , beginning with its definition.

Note that if a maximal finite normal subgroup exists, then it must be unique. Hence “the” unique maximal finite normal subgroup in the definition makes sense.

Perhaps one of the natural questions is the existence of K ⁢ ( G ) . Of course, K ⁢ ( G ) may not exist even if 𝐺 is finitely generated. For a concrete example, note that there is a finitely generated group 𝐺 whose center Z ⁢ ( G ) is isomorphic to the quotient group Q / Z due to Abderezak Ould Houcine [60]. Recall that the center Z ⁢ ( G ) is a characteristic subgroup of 𝐺, and the subgroups ⟨ 1 / p ⟩ < Q / Z are also characteristic subgroups (this follows from the fact that ⟨ 1 / p ⟩ is the unique subgroup with 𝑝 elements). Thus we conclude that the subgroups ⟨ 1 / p ⟩ are finite normal subgroups of 𝐺. Since 𝐺 has subgroups ⟨ 1 / p ⟩ for every prime 𝑝, this implies that 𝐺 does not have the unique maximal finite normal subgroup.

However, it is known that, for any acylindrically hyperbolic group 𝐺, K ⁢ ( G ) exists. We refer to [27, Theorem 2.24].

The amenable radical A ⁢ ( G ) is the unique maximal amenable normal subgroup. For more details about amenability, we refer to [31, Section 18.3] and [50, Chapter 9]. In contrast to K ⁢ ( G ) , it is known that every group has an amenable radical. When 𝐺 is acylindrically hyperbolic, its amenable radical is finite [58, Corollary 7.3], which directly implies K ⁢ ( G ) = A ⁢ ( G ) .

3.2 Description of the kernel of actions on asymptotic cones

Next, we will describe the kernel of group actions on asymptotic cones. Let 𝐺 be a group, and we fix an ultrafilter 𝜔 and sequence d n . The value d ∞ ⁢ ( { x n } , { g ⁢ x n } ) is a metric between { x n } and { g ⁢ x n } in the asymptotic cone. Note that this value is exactly

Here, 𝑆 is a finite generating set for 𝐺. Since this value measures how far an element 𝑔 moves { x n } , the kernel can be expressed as follows:

3.3 Finite conjugacy classes subgroup

Now, we give the definition of FC ⁡ ( G ) . Recall that FC stands for “Finite Conjugacy classes”.

Then it is straightforward that FC ⁡ ( G ) is a normal subgroup of 𝐺.

3.4 Easily obtained inclusions and equalities

Now we will prove some easily obtained equalities and inclusions.

However, these subgroups in Lemma 3.3 may not coincide in general. For the first case, just consider G = Z . In order to find a counterexample for the second case, consider the three-dimensional Heisenberg group 𝐻. Let

be two generators of 𝐻, and let z = [ x , y ] : = x − 1 y − 1 x y . It is known that the commutator subgroup of 𝐻 is an infinite cyclic group generated by 𝑧, and 𝑧 is a distortion element. Here, we say that g ∈ G is a distortion element if 𝑔 has an infinite order and

for some word metric ∥ ⋅ ∥ with respect to some finite generating set. The distortedness can be obtained from [ x l , y m ] = z l ⁢ m .

Now we will give an example of FC ⁡ ( G ) ⊊ ker ⁡ ( G → Isom ⁡ ( Cone ω ⁡ ( G , d n ) ) ) . Let G = H , the Heisenberg group, and pick g = x , one of its generators. Then g ∉ FC ⁡ ( G ) since

Our goal is to show that 𝑔 is in ker ⁡ ( G → Isom ⁡ ( Cone ω ⁡ ( G , d n ) ) ) . First note that any element in 𝐻 can be expressed as

Then we have

Now assume g ∉ ker ⁡ ( G → Isom ⁡ ( Cone ω ⁡ ( G , d n ) ) ) . Then there exists a sequence { x n } in 𝐻 such that lim ω ∥ x n ∥ S d n = L > 0 and { x n } ≠ g ⋅ { x n } , equivalently,

Since 𝑔 is fixed and lim ω d n ∥ x n ∥ S = 1 / L > 0 , we have

which means that

Let x n = y b n ⁢ z c n ⁢ x a n , so

Since 𝑧 is distortion,

and this implies

However, ∥ x n ∥ S ≥ b n , a contradiction. Therefore,

which means FC ⁡ ( G ) ⊊ ker ⁡ ( G ↷ Cone ω ⁡ ( G , d n ) ) .

Also, the following inclusion is known to be satisfied for a countable group. Note that we concentrate only on finitely generated groups, so this inclusion holds in our setting.

This inclusion follows immediately using the AC -center. This notion was first suggested in [74], but we use the definition introduced in [47].

Like the previous inclusions, the two subgroups FC ⁡ ( G ) and A ⁢ ( G ) need not be the same generally. The Heisenberg group 𝐻 also provides a counterexample. Recall that g : = x ∉ FC ( H ) but A ⁢ ( H ) = H since 𝐻 is nilpotent, so FC ⁡ ( H ) ⊊ A ⁢ ( H ) .

The only remaining inclusion relation concerns

In Section 7, we give an example for ker ⁡ ( G ↷ Cone ω ⁡ ( G , d n ) ) ⊊ A ⁢ ( G ) (see Lemma 7.5). However, we have not found a group satisfying the reverse inclusion.

The referee pointed out to the authors that [48] provides examples of finitely generated groups with a trivial amenable radical that are not C ∗ -simple. Furthermore, [14] characterizes C ∗ simplicity (for any discrete group) as the existence of a topologically free boundary action. This suggests that the examples in [48, 14] could be promising candidates for a negative answer to Question 3.6. Thus it would be interesting to compute the kernel of the natural action on the asymptotic cone for these examples.

Recall that the equality among K ⁢ ( G ) , FC ⁡ ( G ) , and A ⁢ ( G ) does not hold in general, but they are the same when 𝐺 is acylindrically hyperbolic.