Transitivity degrees of countable groups and acylindrical hyperbolicity

Transitivity degrees of countable groups and acylindrical hyperbolicity

M. Hull, D. Osin
Abstract

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely and . Here by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group admitting a highly transitive faithful action, we prove the following dichotomy: Either contains a normal subgroup isomorphic to the infinite alternating group or resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.

1 Introduction

Recall that an action of a group on a set is -transitive if and for any two -tuples of distinct elements of , and , there exists such that for . The transitivity degree of a countable group , denoted , is the supremum of all such that admits a -transitive faithful action. For finite groups, this notion is classical and fairly well understood. It is easy to see that , , and it is a consequence of the classification of finite simple groups that any finite group other than or has . Moreover, if is not , , or one of the Mathieu groups , , , , then (see [18]).

For infinite groups, however, very little is known. For example, we do not know the answer to the following basic question: Does there exist an infinite countable group of transitivity degree for every ? There are examples for , and , but the problem seems open even for . There is also a new phenomenon, which does not occur in the finite world: highly transitive actions. Recall that an action of a group is highly transitive if it is -transitive for all . We say that a group is highly transitive if it admits a highly transitive faithful action; it is easy to see that a countably infinite group is highly transitive if and only if it embeds as a dense subgroup in the infinite symmetric group endowed with the topology of pointwise convergence. Obviously whenever is highly transitive, but we do not know if the converse is true. Yet another interesting question is whether there exists a reasonable classification of highly transitive groups (or, more generally, groups of high transitivity degree). The main goal of this paper is to address these questions in certain geometric and algebraic settings.

We begin by discussing known examples of highly transitive groups. Trivial examples of such groups are (the group of all finitary permutations of ) and the infinite alternating group (the group of all finitary permutations that act as even permutations on their supports). Clearly every subgroup of that contains is also highly transitive. A fairly elementary argument allows one to construct finitely generated groups of this sort; we record the following.

Proposition 1.1 (Prop. 5.13).

For every finitely generated infinite group , there exists a finitely generated group such that and . In particular, is highly transitive.

This proposition can be used, for instance, to construct finitely generated highly transitive torsion groups by taking to be finitely generated and torsion. Another interesting particular example of this sort is the family of Houghton groups , , which correspond to the case . These groups are elementary amenable and finitely presented for (see Example 6.6).

On the other hand, there are highly transitive groups of a completely different nature. The first such an example is , the free group of rank . Highly transitive faithful actions of were constructed by McDonough [39], and it was shown by Dixon [17] that in some sense most finitely generated subgroups of are free and highly transitive. Other known examples of highly transitive groups include fundamental groups of closed surfaces of genus at least [35] and, more generally, non-elementary hyperbolic groups without non-trivial finite normal subgroups [14], all free products of non-trivial groups except [22, 26, 28, 42] and many other groups acting on trees [20]. Garion and Glassner [21] used an interesting approach employing Tarski Monster groups constructed by Olshanskii [43] to show that is highly transitive for . Motivated by similarity between and mapping class groups, they ask which mapping class groups are highly transitive. Another question from [21] is whether is highly transitive.

The first main result of our paper is a theorem which simultaneously generalizes most results mentioned in the previous paragraph and allows us to answer the questions of Garion and Glassner. Recall that an isometric action of a group on a metric space is acylindrical if for all , there exist such that for all with , there are at most elements satisfying and . For example, it is easy to show that the action of a fundamental group of a graph of groups with finite edge groups on the associated Bass-Serre tree is acylindrical. A group is called acylindrically hyperbolic if it is not virtually cyclic and admits an acylindrical action on a hyperbolic metric space with unbounded orbits [49]. The class of acylindrically hyperbolic groups includes many examples of interest; for an extensive list we refer to [15, 41, 49, 48].

Every acylindrically hyperbolic group has a maximal normal finite subgroup called the finite radical of and denoted [15].

Theorem 1.2.

Every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. In particular, every countable acylindrically hyperbolic group with trivial finite radical is highly transitive.

It is well-known and easy to see that a group of transitivity degree at least cannot have finite normal subgroups (see Lemma 4.2 (b)). Thus the kernel of a highly transitive action of a countable acylindrically hyperbolic group cannot be smaller than . In particular, the assumption about triviality of finite radical cannot be dropped from the second sentence of the theorem.

Our proof of Theorem 1.2 is based on the notion of a small subgroup of an acylindrically hyperbolic group introduced in Section 2. This notion seems to be of independent interest. In addition, the proof of Theorem 1.2 makes use of hyperbolically embedded subgroups introduced in [15] and small cancellation theory in acylindrically hyperbolic groups developed in [32].

Here we mention just few particular cases of our theorem considered in Corollaries 4.9, 4.14, and 4.194.23. By we denote a times punctured compact orientable surface of genus with boundary components.

Corollary 1.3.
  1. Let be a countable group hyperbolic relative to a collection of proper subgroups. Then is highly transitive if and only if it is not virtually cyclic and has no non-trivial finite normal subgroups.

  2. For , the mapping class group is highly transitive if and only if , , and .

  3. (cf. [21, Theorem 1]) is highly transitive if and only if .

  4. Let be a compact irreducible -manifold. Then is highly transitive if and only if it is not virtually solvable and is not Seifert fibered.

  5. A right angled Artin group is highly transitive if and only if it is non-cyclic and directly indecomposable.

  6. Every -relator group with at least generators is highly transitive.

Recall that a free product of two non-trivial groups has no non-trivial finite normal subgroups. Thus part (a) of Corollary 1.3 covers the case of free products considered in [22, 26, 28] as well as some results of [20]; it also covers the result about hyperbolic groups [14]. Parts (b) and (c) answer the questions of Garion and Glassner mentioned above. Our contribution to (c) is the case .

Theorem 1.2 also allows us to show that in various geometric and algebraic settings, transitivity degree of an infinite group can only take two values, namely and , and infinite transitivity degree is equivalent to being highly transitive. More precisely, we consider three conditions for a group :

  1. .

  2. is highly transitive.

  3. is acylindrically hyperbolic with trivial finite radical.

In Section 4, we show that an infinite subgroup of a mapping class group is either acylindrically hyperbolic or satisfies certain algebraic conditions which imply . A similar result for subgroups of -manifold groups was obtained in [41]. Combining this with Theorem 1.2, we obtain the following.

Corollary 1.4 (Cor. 4.13, Cor. 4.15).

Let be an infinite subgroup of for some or an infinite subgroup of the fundamental group of a compact -manifold. Then and conditions (C)–(C) are equivalent.

A similar result can also be proved in some algebraic settings, e.g., for subgroups of finite graph products. Let be a graph (without loops or multiple edges) with vertex set and let be a family of groups indexed by vertices of . The graph product of with respect to , denoted , is the quotient group of the free product by the relations for all , whenever and are adjacent in . Graph products simultaneously generalize free and direct products of groups. Basic examples are right angled Artin and Coxeter groups, which are graph products of copies of and , respectively. The study of graph products and their subgroups has gained additional importance in view of the recent breakthrough results of Agol, Haglund, Wise, and their co-authors, which show that many groups can be virtually embedded into right angled Artin groups (see [1, 27, 57] and references therein).

Corollary 1.5 (Cor. 4.17).

Let be a countably infinite subgroup of a finite graph product . Suppose that is not isomorphic to a subgroup of one of the multiples. Then and conditions (C)–(C) are equivalent. If, in addition, every is residually finite, then .

To state our next result we need some preparation. Let denote the free group of rank and recall that a group satisfies a mixed identity for some if every homomorphism that is identical on sends to . We say that the mixed identity is non-trivial if as an element of . For the general theory of mixed identities and mixed varieties of groups we refer to [3]. We say that is mixed identity free (or MIF for brevity) if it does not satisfy any non-trivial mixed identity.

The property of being MIF is much stronger than being identity free and imposes strong restrictions on the algebraic structure of . For example, if has a non-trivial center, then it satisfies the non-trivial mixed identity , where . Similarly, it is easy to show that an MIF group has no finite normal subgroups, is directly indecomposable, has infinite girth, etc. (see Proposition 5.4). Other examples of groups satisfying a non-trivial mixed identity are Thompson’s group [55] or any subgroup of containing (see Theorem 1.6 below).

It is also worth noting that MIF groups resemble free products from the model theoretic point of view. More precisely, a countable group is MIF if and only if and are universally equivalent as -groups for all . This means that a universal first order sentence with constants from holds true in if and only if it holds true in . Groups universally equivalent to as -groups are exactly the coordinate groups of irreducible algebraic varieties over [7]. For more details we refer to Section 5 and Proposition 5.3.

Highly MIFtransitive
Figure 1:

In Section 5, we prove the following dichotomy for highly transitive countable groups, which captures the principal difference between examples provided by Proposition 1.1 and Theorem 1.2.

Theorem 1.6.

Let be a highly transitive countable group. Then exactly one of the following two mutually exclusive conditions holds.

  1. contains a normal subgroup isomorphic to .

  2. is MIF (equivalently, and are universally equivalent as -groups for every ).

It is known that infinite normal subgroups of acylindrically hyperbolic groups are also acylindrically hyperbolic and hence cannot be torsion [49]. Thus condition (a) from Theorem 1.6 cannot hold if is acylindrically hyperbolic. Combining this with Theorem 1.2 we obtain the following.

Corollary 1.7 (Cor. 5.10).

Let be an acylindrically hyperbolic group with trivial finite radical. Then is MIF.

Clearly is not MIF whenever . In the particular case of non-cyclic torsion free hyperbolic groups, the above corollary was proved in [2] by different methods.

Theorem 1.2 and Theorem 1.6 are summarized in Fig. 1. Here by we denote the class of acylindrically hyperbolic groups with trivial finite radical. In Section 5 we show that all inclusions are proper. We mention two examples.

Proposition 1.8 (Cor. 5.12, Cor. 5.17).
  1. The group is highly transitive and MIF, but not acylindrically hyperbolic.

  2. There exist finitely generated MIF groups of transitivity degree .

The paper is organized as follows: In Section 2 we review properties of acylindrically hyperbolic groups and hyperbolically embedded subgroups necessary for the proof of Theorem 1.2. In Section 3 we prove Theorem 1.2, and in Section 4 we apply this theorem to various classes of groups to obtain Corollaries 1.3-1.5. In Section 5 we study the relationship between mixed identities and highly transitive actions and prove Theorem 1.6. Some open questions and relevant examples are discussed in Section 6.

Acknowledgments.

We would like to thank Ilya Kapovich, Olga Kharlampovich, and Alexander Olshanskii for helpful discussions of various topics related to this paper. We are especially grateful to Dan Margalit for answering our numerous questions on mapping class groups. The second author was supported by the NSF grant DMS-1308961.

2 Preliminaries on acylindrically hyperbolic groups

2.1. Generating alphabets and Cayley graphs.

When dealing with relative presentations of groups, we often need to represent the same element of a group by several distinct generators. Thus, instead of a generating set of , it is more convenient to work with an alphabet given together with a (not necessarily injective) map such that generates . We begin by formalizing this approach.

Let be a set, which we refer to as an alphabet. Let be a group and let be a (not necessarily injective) map. We say that is generated by (or is a generating alphabet of ) if is generated by . Note that a generating set can be considered as a generating alphabet with the obvious map .

By the Cayley graph of with respect to a generating alphabet , denoted , we mean a graph with vertex set and the set of edges defined as follows. For every and every , there is an oriented edge in labelled by . Given a (combinatorial) path in , we denote by its label. Note that if is not injective, may have multiple edges. Of course, the identity map on induces an isometry between vertex sets of the graphs and . Thus we only need to distinguish between and when dealing with labels; in all purely metric considerations we do not need generating alphabets and can simply work with generating sets.

Given a generating set of and an element , let denote the word length of with respect to , that is the length of a shortest word in that represents . For , we define . Finally, let

2.2. Hyperbolically embedded subgroups.

The typical situation when we apply the language described above is the following. Suppose that we have a group , a subgroup of , and a subset such that and together generate . We think of and as abstract sets and consider the disjoint union

(1)

and the map induced by the obvious maps and . By abuse of notation, we do not distinguish bretween subsets and of and their preimages in . This will not create any problems since the restrictions of on and are injective. Note, however, that is not necessarily injective. Indeed if and subgroup intersect in , then every element of will have at least two preimages in : one in and another in (since we use disjoint union in (1)).

Convention 2.1.

Henceforth we always assume that generating sets and relative generating sets are symmetric. That is, if , then . In particular, every element of can be represented by a word in .

In these settings, we consider the Cayley graphs and and naturally think of the latter as a subgraph of the former. We introduce a relative metric as follows. We say that a path in is admissible if it contains no edges of . Let be the length of a shortest admissible path in that connects to . If no such a path exists, we set . Clearly satisfies the triangle inequality, where addition is extended to in the natural way.

Definition 2.2.

A subgroup of is hyperbolically embedded in with respect to a subset , denoted , if the following conditions hold.

  1. The group is generated by together with and the Cayley graph is hyperbolic.

  2. Any ball (of finite radius) in with respect to the metric contains finitely many elements.

Further we say that is hyperbolically embedded in and write if for some .

Hyperbolically embedded subgroups were introduced and studied in [15] as a generalizaton of relatively hyperbolic groups; indeed, is hyperbolic relative to if and only if with [15, Proposition 4.28]. The following lemma is a particular case of this result.

Lemma 2.3.

For any group , .

In the next two lemmas, we let be a group and a subgroup of .

The following is a particular case of [15, Proposition 4.35].

Lemma 2.4.

Suppose that for some and there is a subset and a subgroup . Then .

Given a group , a subgroup , and , we denote by the conjugate .

Lemma 2.5.

[15, Proposition 2.8] If , then for all , .

2.3. Acylindrically hyperbolic groups.

Recall that an isometric action of a group on a metric space is acylindrical if for every , there exist such that for every two points with , there are at most elements satisfying

Given a group acting on a hyperbolic space , an element is called loxodromic if the map defined by is a quasi-isometry for some (equivalently, any) . Every loxodromic element has exactly limit points on the Gromov boundary . Loxodromic elements are called independent if the sets and are disjoint.

We will often use the following.

Theorem 2.6 ([49, Theorem 1.1]).

Let be a group acting acylindrically on a hyperbolic space. Then exactly one of the following three conditions holds.

  1. is elliptic.

  2. contains a loxodromic element such that .

  3. contains infinitely many loxodromic elements that are independent (i.e., have disjoint limit sets on the boundary).

Definition 2.7.

A group is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space.

In the case of acylindrical actions on hyperbolic spaces being non-elementary is equivalent to the action having unbounded orbits and being not virtually cyclic by theorem 2.6.

It is easy to see from the definition that for every group we have and for every finite subgroup of . Following [15], we call a hyperbolically embedded subgroup non-degenerate if is infinite and .

The next result is a part of [49, Theorem 2.2]. In particular, it allows us to apply all results from [15] concerning groups with non-degenerate hyperbolically embedded subgroups to acylindrically hyperbolic groups.

Theorem 2.8.

A group is acylindrically hyperbolic if and only if contains a non-degenerate hyperbolically embedded subgroup.

If is acylindrically hyperbolic, then has a unique, maximal finite normal subgroup called the finite radical of [15, Theorem 6.14].

Lemma 2.9.

[32] Let be acylindrically hyperbolic. Then is acylindrically hyperbolic and has trivial finite radical.

2.4. Small subgroups in acylindrically hyperbolic groups.

Given a group and a generating set (or an alphabet) , we say that the Cayley graph is acylindrical if so is the action of on . A hyperbolic space is called non-elementary if , where denotes the Gromov boundary of .

Definition 2.10.

We call a subgroup of a group small in (or just small if is understood) if there exists a generating set of such that and is hyperbolic, non-elementary, and acylindrical.

Note that every group containing small subgroups is acylindrically hyperbolic.

Lemma 2.11.

Assume that a group is acylindrically hyperbolic and . Then for every small subgroup , the action of on the coset space is faithful.

Proof.

Let denote the kernel of the action of on . Let be a generating set of such that and is acylindrical, hyperbolic, and non-elementary. Note that acts elliptically on as . Recall that for every acylindrical non-elementary action of a group on a hyperbolic space, every normal elliptic subgroup is finite [49, Lemma 7.2]. Applying this to the group acting on , we obtain that and consequently . ∎

The proof of Theorem 1.2 will make use of one particular example of small subgroups. Recall that given a generating set of a group , and denote the corresponding word length and metric on , respectively.

Proposition 2.12.

Let be a group, a subgroup of , a subset of such that . Suppose that a subgroup is generated by a subset such that

and

for all . Then is small. Moreover, for every subgroup that properly contains , there exists a subset such that is hyperbolic and acylindrical, , and the action of on is non-elementary.

The proof can be easily extracted from [49]. In order to make our paper self-contained, we recall necessary definitions and results from [49].

Let and let be a (combinatorial) path in . A non-trivial subpath of is called an -component if is a word in the alphabet and is not contained in any bigger subpath of with this property. If is an -component of and are endpoints of , then all vertices of belong to the same coset ; in this case we say that penetrates and call the number the depth of the penetration. Given two elements , we say that a coset is -separating if there exists a geodesic from to in such that penetrates and the depth of the penetration is greater than .

The term “separating” is justified by the following result, which is a simplification of [49, Lemma 4.5].

Lemma 2.13.

Let be a group, a subgroup of , a subset of such that . Then for any constants and , there exists such that the following holds. Let and let be an -separating coset. Then every -quasi-geodesic in that connects to penetrates .

It is observed in [49] that if , then the Cayley graph is not necessarily acylindrical. However, the following lemma holds true. Its first claim is a particular case of Theorem 5.4 in [49]; the second claim is not stated in that theorem itself, but is obvious from its proof.

Lemma 2.14.

Let be a group, a subgroup of , a subset of such that . Then there exists a set such that and is acylindrical. More precisely, given a constant , let denote the set of all elements such that the set of all -separating cosets is empty. Then for every large enough , the set satisfies the above requirements.

We are now ready to prove Proposition 2.12.

Proof.

Let . For every element , let be a shortest path in connecting to such that is a word in the alphabet . In particular, does not penetrate any left coset of . It is straightforward to check that is -quasi-geodesic for some and that only depend on and (in fact, we can take and ).

Let be a constant such that the conclusion of Lemma 2.14 holds and , where is given by Lemma 2.13. Then for any , there are no -separating cosets. Therefore, the set defined in Lemma 2.14 contains . By Lemma 2.14, is acylindrical.

It remains to show that the action of on is non-elementary. By Theorem 2.6, it suffices to show that has unbounded orbits and is not virtually cyclic. Let . By [15, Corollary 6.12], there exists such that is loxodromic with respect to the action on . Since , we obtain that the orbits of are unbounded. Finally assume that is elementary. Since is infinite, it must be of finite index in . In particular, contains an infinite subgroup such that . Then for every , we obtain which contradicts Lemma 2.5 if . Thus the action of on is non-elementary. ∎

3 Highly transitive actions of acylindrically hyperbolic groups

3.1. Outline of the proof.

The goal of this section is to prove Theorem 1.2. We begin with a brief sketch of the proof.

We first note that by Lemma 2.9, it suffices to deal with the case of trivial finite radical. Assuming , we will construct a subgroup such that the action of on the left cosets of will be highly transitive. will be defined as the union of a sequence of subgroups

Given and two tuples of pairwise distinct cosets and , we define by choosing an element and setting

(2)

Then clearly , so after enumerating all tuples of elements of and repeating this process inductively we will get that the action of on the set of left cosets of is highly transitive, provided . To the best of our knowledge, this natural construction first appeared in [4]. It was also used by Chaynikov [14] to prove Theorem 1.2 in the particular case when is hyperbolic.

The main difficulty is to ensure that that the action of on is faithful. Note that if the elements is not chosen carefully, then we are likely get . In Chainikov’s paper [14], the proof of faithfulness of the action is built on the well-developed theory of quasi-convex subgroups of hyperbolic groups. Currently there is no analogue of this theory in the more general context of acylindrically hyperbolic groups, so we choose another approach based on the notion of a small subgroup introduced in the previous section and small cancellation theory in acylindrically hyperbolic groups developed in [32].

More precisely, we prove in Lemma 3.1 that if is small for some , then by choosing the element carefully (small cancellation theory is employed here) one can ensure that the subgroup defined by (2) is still small. In addition, we show that is is any finite subset of disjoint from , we can keep it disjoint from . Recall also that the action of on the space of left cosets is faithful whenever is a small subgroup and (Lemma 2.11). These results allow us to iterate the above construction in such a way that the resulting action of on is faithful.

3.2. The inductive step.

The main goal of this subsection is to prove the following proposition, which takes care of the inductive step in the proof of Theorem 1.2. Given a group , a subgroup , and two collections of elements and of , we say that the triple is admissible if

(3)

for all .

Proposition 3.1.

Let be a group, a small subgroup of , and elements of . Assume that the triple is admissible. Let also be a finite subset disjoint from . Then there exists and a subgroup such that

  1. .

  2. is small in .

  3. for all .

  4. .

We first recall some well-known results and terminology concerning free products.

Theorem 3.2.

[36, Chapt IV, Theorem 1.2] Let , be arbitrary groups. Each element of a free product can be uniquely expressed as where , each is a non-trivial element of one of the factors and successive elements belong to different factors.

The expression as in the above theorem representing an element is called the normal form of the element . The following corollary is an immediate consequence of the uniqueness of normal forms.

Corollary 3.3.

Let and let be a generating set for . Let be the normal form for , where . Then

In particular, the embeddings of metric spaces are isometric.

Lemma 3.4.

Let be a group with a generating set , a subgroup of such that . Let , be elements of . Assume that the triple is admissible and let ,

and

Then and the natural map is Lipschitz.

Proof.

First, if , then . Assume now that . Let

for and let be a minimal word in representing . Then

for some , where each is a (possibly trivial) element of and . Then , and note that each . Let , where

for and

for . We also define . Then

(4)

We show now that consecutive -letters in this expression do not cancel, which means that (4) becomes a normal form for in the free product after possibly removing the trivial and combining adjacent powers of . Suppose for some , and . Then . If , then since . If , then , for otherwise and would freely cancel in the word . Hence since it is a conjugate of in this case. By a similar argument, it follows that if and , then . Thus, if , then and hence the corresponding -letters cannot cancel with each other.

Since the normal form for contains at least one -letter, Theorem 3.2 gives that and hence . Furthermore, by Corollary 3.3,

(5)

Since each , it follows that

By definition, a group is acylindrically hyperbolic if it admits a non-elementary acylindrical action on some hyperbolic metric space. In [49] it is shown that an acylindrically hyperbolical group G always has such an action on a hyperbolic space of the form where is a generating set of .

Definition 3.5.

Let be a generating set of such that is hyperbolic and acylindrical. A subgroup is called suitable with respect to A if the action of on is non-elementary and does not normalize any non-trivial finite subgroup of .

The following is a modification of [32, Theorem 7.1]; note that condition (a) is a simplification of [32, Theorem 7.1 (d)] and condition (b) is an immediate consequence of the proof of [32, Theorem 7.1]. Recall that denotes the ball of radius in with respect to the metric .

Theorem 3.6.

Let be a generating set of such that is hyperbolic and acylindrical, and let be suitable with respect to . Let and . Then there exists a group and a surjective homomorphism such that

  1. There exists a generating set of such that and is hyperbolic, acylindrical, and non-elementary.

  2. is injective.

  3. for some . In particular, .

We are now ready to prove the main result of this subsection.

Proof of Proposition 3.1.

Since is small, there exists a generating set such that and is hyperbolic, acylindrical, and non-elementary. By Theorem 2.6, there exists loxodromic with respect to the action on . Then ([32, Corollary 3.16]) and hence by Lemma 2.3 and Lemma 2.4.

Let and let , , and be as in Lemma 3.4. Obviously is bounded with respect to the metric . By Lemmas 3.4 and 2.12 applied to the group , hyperbolically embedded subgroup , generating set , and subgroups and , we can chose a generating set of such that , is acylindrical, hyperbolic, and the action of on is non-elementary.

Since , does not normalize any non-trivial finite normal subgroup of and hence is suitable with respect to . Since is finite, we can choose such that . Applying Theorem 3.6 gives a group and a surjective homomorphism . By condition (c) of Theorem 3.6 and the obvious Tietze transformation, . In particular, the composition of the natural embedding and is the identity map on . Let and . Since , . By construction for , so the definition of and gives that for . Since , and condition (a) of Theorem 3.6 gives that is small in . Since and is disjoint from , is also disjoint from by Lemma 3.4. Finally, , so condition (c) of Theorem 3.6 gives that . ∎

3.3. Proof of Theorem 1.2.

Below we employ the standard notation for elements of a group.

Proof.

By Lemma 2.9 it suffices to prove the theorem in the case . Let

If , , and , we write to mean for all . We also enumerate all elements of :

We will construct a sequence of subgroups of and a sequence of elements of such that the following conditions hold for every .

  1. is small in .

  2. For each , we have .

  3. If and the triple is admissible, then there exists such that .

We proceed by induction. Set . Assume now that we have already constructed subgroups satisfying (a)–(c) for some .

The action of on the coset space is faithful by (a) and Lemma 2.11. Hence there exists such that . Let

Note that is disjoint from by part (b) of the inductive assumption and the choice of .

Let and . If the triple is not admissible, we set . All inductive assumptions obviously hold in this case. Henceforth we assume that is admissible. Then we can apply Proposition 3.1 to the small subgroup and let . Conditions (a)–(c) for follow immediately from the proposition.

Let , and let denote the corresponding coset space. We first notice that the action of on is faithful. Indeed assume that some element acts trivially. Then all conjugates of by elements of belong to . In particular, and hence for all sufficiently large . However this contradicts (b).

Further, for any two collections of pairwise distinct cosets and , there exists such that , . Since , the triple is admissible. Then by (c) we have for all , which obviously implies . Thus the action of on is highly transitive. ∎

4 Transitivity degrees of mapping class groups, -manifold groups, and graph products

4.1. Preliminaries on permutation groups.

We begin by reviewing some basic concepts and facts about multiply transitive groups used in this section. For more details we refer the reader to [16, 18].

By “countable” we mean finite or countably infinite. Let denote a countable set and let denote the symmetric group on . We will also use the notation for the symmetric group on a countable infinite set. A permutation group is -transitive if the action of on is -transitive; is primitive if it is transitive and does not preserve any non-trivial equivalence relation (or partition) on . Here an equivalence relation on is called trivial, if it is equality or if all elements of are equivalent.

In the following two lemmas, we collect some well-known properties of primitive and -transitive groups which will be used many times throughout this section. Since the proofs are short and fairly elementary, we provide them for convenience of the reader.

Lemma 4.1.

Let be a primitive permutation group.

  1. Every non-trivial normal subgroup of is transitive. In particular, if is infinite then it has no non-trivial finite normal subgroups.

  2. For every , is a maximal subgroup of .

Proof.

Let be primitive and . The collection of -orbits defines a -invariant partition of , hence by primitivity this partition is trivial. This means that either acts transitively on or . Further, let . Then the -translates of the orbit form a -invariant partition of . By primitivity either or . In the former case we have ; in the latter case . ∎

Recall that a group has infinite conjugacy classes, abbreviated ICC, if every nontrivial conjugacy class of is infinite. Note that ICC groups cannot have non-trivial center or non-trivial finite normal subgroups.

Lemma 4.2.

Let be a -transitive permutation group.

  1. is primitive.

  2. If is infinite, then it is ICC.

Proof.

Part (a) is obvious. To prove (b), let and let be distinct elements of . (Note that is infinite since so is .) Let also be an element of such that . Since is 2-transitive, for all there exists such that and . Then ,which means that are all distinct conjugates of . Hence is ICC.

The next lemma is also quite elementary. It will be used to bound transitivity degree of various groups from above.

Lemma 4.3.

Let be an infinite primitive permutation group, a transitive subgroup of , and a non-trivial normal subgroup of . Suppose that . Then is an infinite minimal normal subgroup of .

Proof.

Let be a non-trivial subgroup of such that . Consider any and fix some . By part (a) of Lemma 4.1, is transitive. Hence there exists such that . That is, for . Since is transitive, for every there is such that . Since we obtain

Thus fixes pointwise, i.e., , which in turn implies . As this is true for every , we obtain