Tits alternatives for graph products

Tits alternatives for graph products

Abstract.

We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank . In the course of the paper we develop the theory of parabolic subgroups, which allows to describe the structure of subgroups of graph products that contain no non-abelian free subgroups. We also obtain a number of results regarding the stability of some group properties under taking graph products.

Key words and phrases:
Graph products, Tits Alternative, right angled Coxeter groups, right angled Artin groups, graph groups.
2010 Mathematics Subject Classification:
Primary 20F65, secondary 20E07.
This work was supported by the EPSRC grant EP/H032428/1. The first author was also supported by MCI (Spain) through project MTM2008–1550.

Yago Antolín]yago.anpi@gmail.com

Ashot Minasyan]aminasyan@gmail.com

1. Introduction

In 1972 J. Tits [35] proved that a finitely generated linear group either is virtually solvable, or contains a copy of the free group of rank . Nowadays such a dichotomy is called the Tits Alternative. This alternative is a powerful result with important consequences. It was used by M. Gromov in his proof of the famous “polynomial growth” theorem [14]; it also shows that linear groups satisfy von Neumann conjecture: every linear group is either amenable or contains a non-abelian free subgroup.

The Tits Alternative can be naturally modified by substituting “finitely generated”, “virtually solvable” and “contains a non-abelian free subgroup” with other conditions of a similar form. An example of a result of this type is the theorem of G. Noskov and E. Vinberg [30] claiming that every subgroup of a finitely generated Coxeter group is either virtually abelian or large. Recall that a group is said to be large if it has a finite index subgroup which maps onto a non-abelian free group. A large group always contains a non-abelian free subgroup, but not vice-versa.

Let us now formally describe the possible versions of the Tits alternative that we are going to consider. Let be a collection of cardinals; a group is said to be -generated, if there is a generating set of and such that

Definition 1.1.

Suppose that is a collection of cardinals, is a class of groups and is a group. We will say that satisfies the Tits Alternative relative to if for any -generated subgroup either or contains a non-abelian free subgroup.

The group satisfies the Strong Tits Alternative relative to if for any -generated subgroup either or is large.

In this terminology the theorem of Tits [35], mentioned above, tells us that linear groups satisfy the Tits Alternative relative to where is the collection of all finite cardinals and is the class of virtually solvable groups. For the class , of virtually abelian groups, the Tits Alternative relative to is known to hold in any word hyperbolic group [15], [6], where is the free group of rank , and in any group acting freely and properly on a finite dimensional CAT() cubical complex [32].

Let be the collection of all countable cardinals. The result of G. Noskov and E. Vinberg [30] in this language becomes: finitely generated Coxeter groups satisfy the Strong Tits Alternative relative to .

The goal of this paper is to prove that many forms of Tits Alternative are stable under graph products. Let be a simplicial graph and suppose that is a collection of groups (called vertex groups). The graph product , of this collection of groups with respect to , is the group obtained from the free product of the , , by adding the relations

The graph product of groups is a natural group-theoretic construction generalizing free products (when has no edges) and direct products (when is a complete graph) of groups , . Graph products were first introduced and studied by E. Green in her Ph.D. thesis [13]. Further properties of graph products have been investigated by S. Hermiller and J. Meier in [20] and by T. Hsu and D. Wise in [22].

Basic examples of graph products are right angled Artin groups, also called graph groups (when all vertex groups are infinite cyclic), and right angled Coxeter groups (when all vertex groups are cyclic of order ).

Throughout this paper will denote the group of integers under addition and will denote the infinite dihedral group. We will be interested in collections of cardinals and classes of groups satisfying the following properties:

  • (closed under isomorphisms) if are groups, and then ;

  • (closed under -generated subgroups) if and is an -generated subgroup, then ;

  • (closed under direct products of -generated groups) if are -generated then ;

  • (contains the infinite cyclic group) ;

  • (contains the infinite dihedral group) if then .

Theorem A.

Suppose that is a collection of cardinals and is a class of groups enjoying (P)–(P). Let be a finite graph and let be a family of groups. Then the graph product satisfies the Tits Alternative relative to provided each vertex group , , satisfies this alternative.

To establish the strong version of Tits alternative we will need one more condition on and :

  • (-locally profi) if is non-trivial and -generated then possesses a proper finite index subgroup.

Theorem B.

Let be a collection of cardinals and let be a class of groups enjoying the properties (P)-(P), such that contains all finite cardinals or at least one infinite cardinal. Suppose that is a finite graph and is a family of groups. Then the graph product satisfies the Strong Tits Alternative relative to provided each vertex group , , satisfies this alternative.

Examples of classes of groups with properties (P)-(P), for , are the classes consisting of virtually abelian groups, virtually nilpotent groups, (virtually) polycyclic groups, (virtually) solvable groups and, more generally, elementary amenable groups. It is easy to see that all of these properties are necessary. For example, if groups from contain no free subgroups and is a non-trivial group without proper finite index subgroups then will possess no non-trivial finite quotients. It follows that cannot be large; on the other hand, as it contains a copy of . Thus we see that property (P) is necessary for the claim of Theorem B.

For any , let be the class of all solvable groups of derived length at most . Denote by the class of all groups that are virtually in . It is easy to see that the pairs and enjoy the properties (P)-(P), for all with (when the class , of abelian groups, does not satisfy (P)). Hence applying Theorem B we achieve

Corollary 1.2.

Suppose that for some or for some . Let be a graph product of groups from . Then any finitely generated subgroup of either is large or belongs to .

Observe that the graph product could be taken over an infinite graph in the above corollary because a finitely generated subgroup is always contained in a “sub-graph product” with only finitely many vertices (see Remark 3.1). It is also worth noting that the group , of rational numbers, can be embedded into a finitely generated group from (this follows from the fact that there are finitely generated center-by-metabelian groups whose center is the free abelian group of countably infinite rank – see [19]; taking a quotient of such a group by an appropriate central subgroup produces an embedding of an arbitrary countable abelian group into the center of a finitely generated group from ). Consequently, the free square of a finitely generated group from can contain , which is neither solvable nor large. Thus the assumption that is finitely generated in Corollary 1.2 is essential.

Applying Theorem B to finitely generated right angled Coxeter groups, we see that these groups satisfy the Strong Tits Alternative relative to , where is the smallest class of groups containing , closed under isomorphisms, taking subgroups and direct products. It is easy to see that , where denotes the class of polycyclic groups; in particular all groups in will be finitely generated. A theorem of F. Haglund and D. Wise [16] states that every finitely generated Coxeter group is virtually a subgroup of some finitely generated right angled Coxeter group. Therefore we recover the result of Noskov-Vinberg [30], mentioned above:

Corollary 1.3.

Let be a finitely generated Coxeter group. If is an arbitrary subgroup then either is large or is finitely generated and virtually abelian.

Finally we would like to suggest the strongest possible (in our opinion) form of Tits alternative as follows.

Definition 1.4.

Suppose that is a collection of cardinals, is a class of groups and is a group. We will say that satisfies the Strongest Tits Alternative relative to if for any -generated subgroup either or has an epimorphism onto the free group of rank .

The group is an example of a torsion-free large group ([4]) which does not map onto ([27]). Thus the Strongest Tits Alternative is indeed more restrictive than the Strong Tits Alternative. The evident groups, satisfying the Strongest Tits Alternative relative to the class of torsion-free abelian groups (and arbitrary ), are residually free groups.

For a collection of cardinals , we define a new collection of cardinals , by saying that a cardinal belongs to if and only if (see Sub-section 2.1 for the definition of the addition of cardinals). For instance, if then ; note also that and . In order to prove the strongest alternative, we need an additional property of the pair :

  • (-locally indicable) if is non-trivial and -generated then has an infinite cyclic quotient.

Evidently (P) implies that every group in is torsion-free (provided contains at least one non-zero cardinal). Basic examples of groups satisfying (P) with are torsion-free nilpotent groups (see [31, 5.2.20]).

Theorem C.

Let be a collection of cardinals and let be a class of groups enjoying the properties (P)-(P) and (P). Suppose that is a finite graph and is a collection of groups. Then the graph product satisfies the Strongest Tits Alternative relative to provided each vertex group , , satisfies this alternative.

Consider any pair satisfying the conditions of Theorem C, such that . Then for an arbitrary non-trivial -generated group , is -generated and contains a copy of , hence . So, the Strongest Tits Alternative (relative to ) for would imply that maps onto , hence the image of under this homomorphism is non-trivial, and, so it possesses an epimorphism onto . Thus the assumption of (P) in Theorem C is indeed necessary.

For the first application of Theorem C, let us take and let be any class of torsion-free groups enjoying the properties (P)-(P) (e.g., could be the class of torsion-free amenable groups). In this case -local indicability is equivalent to torsion-freeness, and so (P) holds automatically. Recalling that a -generated group maps onto if and only if it is isomorphic to , we achieve

Corollary 1.5.

Let be a non-empty class of torsion-free groups, closed under isomorphisms, direct products and taking subgroups. Suppose that is a graph product with all vertex groups from . Then for any -generated subgroup , either or .

Corollary 1.5 generalizes a classical result of A. Baudisch [3], who proved that any two non-commuting elements of a right angled Artin group generate a copy of . In fact, we are now able to say much more about subgroups of right angled Artin groups. Indeed, taking and to be the class of finitely generated torsion-free abelian groups, and applying Theorem C we obtain the following result:

Corollary 1.6.

Any subgroup of a finitely generated right angled Artin group is either free abelian of finite rank or maps onto .

The finite generation assumption in the above statement is not very important. Indeed, using Corollary 1.6 together with Remark 3.1, it is easy to show that any non-abelian subgroup of an arbitrary right angled Artin group maps onto . An easy consequence of Corollary 1.6 is that for every non-cyclic subgroup , of a right angled Artin group, the first Betti number (i.e, the -rank of the abelianization) is at least two.

As far as we know, Corollary 1.6 gives the first non-trivial family of groups satisfying the Strongest Tits Alternative (it is not hard to construct right angled Artin groups that are not residually free). The family of subgroups of right angled Artin groups is very rich and includes a lot of examples examples (see [1, 5, 10, 16, 17, 26, 37]). However, Corollary 1.6 can be used to show that many groups are not embeddable into right angled Artin groups. More precisely, one can obtain information regarding solutions of an equation in a right angled Artin group from the solutions of the same equation in free and free abelian groups. For example, we have

Corollary 1.7.

If three elements of a right angled Artin group satisfy for , then these elements pairwise commute.

The above fact is an immediate consequence of Corollary 1.6 and the result of R. Lyndon and M. Schützenberger [29], who proved the same statement when is free. Corollary 1.7 generalizes a theorem of J. Crisp and B. Wiest [10, Thm. 7], who established its claim in the case when .

The paper is organized as follows. In Section 2 we recall some basic properties of graph products, and in Section 3 we develop the theory of parabolic subgroups of graph products. The results of this section, together with Bass-Serre theory, allow us to prove, in Section 4, the Structure Theorem for subgroups of graph products which do not contain a copy of (Theorem 4.1). The Structure Theorem is one of the main results of this work, and it is intensively used in the proofs of Theorems A, B and C. We apply the Structure Theorem to give a more detailed description of “small” subgroups of graph products (see Theorem 4.3), and Theorem A is a consequence of this description.

In order to prove Theorems B and C, we study the kernel of the canonical retraction of a graph product onto one of the vertex groups and prove that this kernel is again a graph product (Theorem 5.2). This fact has a number of interesting applications, which are established in Section 5. In Section 6 we establish a sufficient criterion for a subgroup of a graph product to be large or to map onto (Theorem 6.5), and then we prove Theorems B and C. Finally, in Section 7 we construct two examples showing that the claims of Corollaries 1.5 and 1.6 are in a certain sense optimal. Example 7.1 shows that without the assumption of torsion-freeness, there is no control over -generated subgroups of graph products; and Proposition 7.3 demonstrates that there is no connection between the rank of a non-abelian subgroup of a right angled Artin group and the maximal rank of the free group onto which it can be mapped. The latter is based on a Rips-type construction, which is presented in Proposition 7.2.

2. Preliminaries

2.1. Cardinal numbers

Our reference for cardinal numbers is [36]. Cardinals are traditionally well-ordered by injection ( if there is an injective map from to ), but, assuming the Axiom of Choice, this ordering is equivalent to the ordering defined via surjections, which we will use (see [36, Proposition 2.1.8]). Formally, if , are two cardinals, and , are sets with , , then if and only if there exists a surjective map The addition of cardinals and is defined in a natural way: is the cardinal of the union of two disjoint sets and with and . A well-known fact tells us that if at least one of the cardinals , is infinite (see [36, Corollary 7.6.2]). If and is a cardinality of a set then denotes the cardinality of the disjoint union of copies of ; thus if is an infinite cardinal then for any .

Let be a collection of cardinals. We say that a group is -generated, if there is a generating set of of cardinality , and a cardinal such that . Remark that the image of an -generated group under a homomorphism is again -generated. It also follows from the definition that if is another collection of cardinals, such that for every there is with , then any -generated group is also -generated. We will say that a collection of cardinals is ample if it contains all finite cardinals or if it contains at least one infinite cardinal.

Lemma 2.1.

Let be an ample collection of cardinals. If is an -generated group and is a subgroup of finite index then is also -generated.

Proof.

By the assumptions, for some , where . Choose a generating set of with for some . By a theorem of Reidemeister and Schreier (see [31, 6.1.8]), can be generated by a subset of the set , where . Consequently . If is finite then is also finite, and hence is -generated by the assumptions on . On the other hand, if is infinite, then is an infinite cardinal, and so , as required. ∎

2.2. Graph products

Let be a graph without loops or multiple edges. We will use and to denote the set of vertices and the set of edges of respectively. An edge can be considered as a -element subset of . A path in of length from to is a sequence of vertices where , for .

For any subset , by we will denote the full subgraph of with vertex set That is, and if and only if .

The link , of a vertex , is the subset of vertices adjacent to (excluding the itself); in other words, For a subset , we define .

Let be a family of groups and let be the corresponding graph product. Any element may be represented as a word where each , called a syllable of , is an element of some and . The number of syllables is the length of the word. For , we will say that the syllables and can be joined together if for some and , for . In this case, in the word represents the same element as the words and , whose lengths are strictly smaller than the length of .

A word is reduced either if it is empty or if for all , and no two distinct syllables of can be joined together.

We define the following transformations for the word .

  1. Remove a syllable if in .

  2. Replace two consecutive syllables and in the same vertex group with the single syllable .

  3. (Syllable shuffling) For consecutive syllables with an edge of interchange and

Note that transformation (T3) preserves the length of the word, and transformations (T1),(T2) decrease it by . Evidently starting with some word and applying finitely many transformations (T1)-(T3) we can obtain a reduced word , representing the same element of the group .

The following theorem was first proved by E. Green [13, Thm. 3.9] in her thesis.

Theorem 2.2 (The Normal Form Theorem).

Every element can be represented by a reduced word. Moreover, if two reduced words represent the same element of the group, then one can be obtained from the other after applying a finite sequence of syllable shuffling. In particular, the length of a reduced word is minimal among all words representing , and a reduced word represents the identity if and only if it is the empty word.

Let and be a reduced word representing We define the length of in to be and the support of in to be By the Normal Form Theorem, the length and support of an element are well defined. For a subset , the support of , will be defined by . Observe that for a subgroup one has . In particular, if is finitely generated then is a finite subset of .

For every , Theorem 2.2 also allows to define the set , consisting of all such that some reduced word , representing , starts with a syllable from . Similarly, will denote the subset of all such that some reduced word for ends with a syllable from . Evidently .

3. Parabolic subgroups of graph products

Throughout this section will denote a simplicial graph, , will be a family of groups and will be the corresponding graph product. For any subset the subgroup , generated by , is called a full subgroup; according to a standard convention, . By the Normal Form Theorem, is the graph product of the groups with respect to the graph . It is also easy to see that there is a canonical retraction defined (on the generators of ) by for each with , and for each with .

Remark 3.1.

If is a finitely generated subgroup then is contained in the full subgroup , of , where is a finite subset of , and thus is a graph products of the family over a finite graph .

For any the group naturally splits as a free amalgamated product: , where , and (cf. [13, Lemma 3.20]).

The goal of this section is to develop the theory of parabolic subgroups (i.e., conjugates of full subgroups) of graph products. Some of the statements in this section are similar to those already known about parabolic subgroups of Coxeter groups (see [25, Sections 2,3]) and about parabolic subgroups of right angled Artin groups (see [12, Section 2]).

Lemma 3.2.

Suppose that and are some elements of such that , and . Then can be represented by a reduced word , where and , with .

Proof.

Choose some reduced words , and representing the elements , , and in respectively. The statement will be proved by induction on . If then the claim clearly holds, so assume that and the claim has already been established for all , satisfying the assumptions of the lemma, with .

If , there is nothing to prove. Otherwise, the equality shows that , representing the left-hand side, cannot be a reduced word. Since the word is reduced and , there must exist such that can be joined with in . It follows that because otherwise the word would not be reduced. It also follows that is represented by the word and , where . Hence we have , where is the element represented by the word . Note that the word is reduced as it was obtained from a reduced word for by removing its last syllable; in particular, .

Suppose that there is some . Then for some , , with and . Observe that must commute with , as , therefore , implying that , which contradicts to our assumptions. Thus and we can apply the induction hypothesis to conclude that is represented by a reduced word , where , and . After setting , we can conclude that is represented by the word (which must be reduced as its length is ), satisfying the required properties. ∎

Lemma 3.3.

Let be an arbitrary collection of subsets of . Then , where .

Proof.

Evidently, . To show the reverse inclusion, consider any . By Theorem 2.2, for every , hence , thus . ∎

Proposition 3.4.

Consider arbitrary and . Then there exist and such that .

Proof.

Obviously one can write , where , and . Then and . If for some with , then evidently , that is can be shortened. Thus without loss of generality we can assume that .

For every element , take some such that . Applying Lemma 3.2 we see that . The latter implies that , i.e., every syllable of commutes with every syllable of . Setting we see that

By Lemma 3.3, , where , and so

as claimed. ∎

Definition 3.5.

Any subgroup , conjugate to a full subgroup, is called parabolic. Moreover, if there are and such that , then is said to be a proper parabolic subgroup. In the latter case unless for all . Clearly any parabolic subgroup , where and , is a retract of , with the retraction , defined by for all .

Proposition 3.4 immediately yields

Corollary 3.6.

The intersection of two parabolic subgroups of a graph product is again a parabolic subgroup.

Lemma 3.7.

If and satisfy then . If, in addition, then there is such that .

Proof.

Denote and let be the retraction defined above. By definition, , and as , we have . Therefore , and . It follows that and .

Now, if , we have . Hence , where . ∎

Corollary 3.8.

If and are such that and for all then .

Lemma 3.9.

If is a parabolic subgroup of a graph product and for some , then .

Proof.

Let be a retraction of onto . For any , take . Then , hence for all , thus . ∎

Proposition 3.10.

Let be a subset of the graph product such that at least one of the following conditions holds:

  • the graph is finite;

  • the subgroup is finitely generated.

Then there exists a unique minimal parabolic subgroup of containing .

Proof.

The uniqueness is clear from Corollary 3.6.

To establish the existence, suppose, at first, that is finite. Then there is a subset , of minimal cardinality, such that for some . Let us show that is a minimal parabolic subgroup containing . Suppose that there is some parabolic subgroup of such that . According to Lemma 3.7, , and the minimality of implies that . Consequently, and . Therefore, applying Lemma 3.9, we can conclude that , as required.

In the case (ii), when for some finite subset , it is clear that , where is a finite subset of . Recall that the full subgroup , of , is itself a graph product (of the groups with respect to the graph ) and any parabolic subgroup of is a parabolic subgroup of . Since , by the first part of the proof is contained in a minimal parabolic subgroup of for some and . If for some and , then and so, by Lemma 3.7, is a parabolic subgroup of . Therefore, using minimality of , we achieve ; thus the proposition is proved. ∎

Definition 3.11.

Suppose that a subset is contained in a minimal parabolic subgroup of . Then this parabolic subgroup will be called the parabolic closure of and will be denoted by .

Recall that the normalizer , of a subset , is the subgroup of defined by .

Lemma 3.12.

Let and . Suppose that the parabolic closure of in exists. Then for any with one has , where . In particular, , i.e., any element normalizing also normalizes .

Proof.

Clearly for any , with , is contained in , which is also a parabolic subgroup. Hence , and Lemma 3.9 implies that , thus . ∎

Proposition 3.13.

Let be a non-trivial parabolic subgroup of the graph product . Choose and so that and for all . Then ; in particular the normalizer is a parabolic subgroup of .

Proof.

After conjugating everything by , we can assume that . Observe that since . Consider any element , any and any . Then for some and Lemma 3.2 implies that . Since the latter holds for every , we see that , where by Lemma 3.3. Thus , and evidently , hence . ∎

For any graph one can define its complement graph to be the graph with the same vertex set such that the edge set is the complement of in the set of two-element subsets of ; in other words, for any , if and only if .

Definition 3.14.

A graph will be called irreducible if cannot be represented as a union of two disjoint non-empty subsets such that .

It is easy to see that is irreducible if and only if is connected. In the case when is not irreducible, any graph product , with respect to , naturally splits as a direct product , where and .

Corollary 3.15.

Assume that is a finite irreducible graph and is the graph product of a family of groups with respect to . Suppose that is a non-trivial subgroup such that is a proper parabolic subgroup of . Then is contained in a proper parabolic subgroup of .

Proof.

Combining Proposition 3.10 with Lemma 3.12 we see that , where . By the assumptions, for some and ; after discarding all with trivial vertex groups, we can suppose that for each . Then , according to Proposition 3.13. Since (as ), we can conclude that is a proper parabolic subgroup of because as is irreducible. ∎

Lemma 3.16.

If is a finite irreducible graph and is a virtually cyclic subgroup which is not contained in any proper parabolic subgroup of , then is finite for each proper parabolic subgroup of .

Proof.

Suppose that is infinite for some proper parabolic subgroup . Then there is an infinite cyclic subgroup such that . Observe that is a proper parabolic subgroup and , therefore, must also be contained in a proper parabolic subgroup by Corollary 3.15, contradicting to our assumptions. Hence . ∎

Proposition 3.17.

Suppose that the graph is finite and . Then there is a finite subset such that