Finite index subgroups of fully residually free groups
Using graph-theoretic techniques for f.g. subgroups of we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. Also we obtain an analogue of Greenberg-Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. non-abelian subgroup of a f.g. fully residually free group is of finite index in its commensurator.
- 1 Introduction
- 2 Preliminaries
- 3 Finite index criteria
- 4 Greenberg-Stallings Theorem
- 5 Commensurator
Fully residually free (or freely discriminated , or -residually free , or limit [15, 16]) groups have been extensively studied over the last ten years. Although appeared first in 60’s (see ) this class of groups drew much attention because of its connection with equations over free groups. Recall that a group is called fully residually free if for any finitely many non-trivial elements there exists a homomorphism of into a free group , such that for . There are other definitions of these groups more or less convenient depending on the setting.
This class of groups can be studied from several viewpoints using many different techniques. In particular, f.g. fully residually free groups are fundamental groups of graphs of groups of a very particular type and their structure can be described using Bass-Serre theory (see [10, 6, 7]). These groups are relatively hyperbolic with respect to their maximal abelian subgroups (see ), which provides another tool of studying them from a geometric viewpoint. It is known that f.g. fully residually free groups these groups act freely on -trees (see ), etc.
Our study of fully residually free groups relies heavily on the fact proved by Kharlampovich and Myasnikov (see ) that every finitely generated fully residually free group is embeddable into , the free exponential group over the ring of integer polynomials . This group was introduced by Lyndon (see ) and he proved that (and hence its subgroups) is fully residually free. It follows that one way to understand the properties of these groups is to study finitely generated subgroups of .
A new technique to deal with became available recently when Myasnikov, Remeslennikov, and Serbin showed that elements of this group can be viewed as reduced infinite words in the generators of (see ). It turned out that many algorithmic problems for finitely generated fully residually free groups can be solved by the same methods as in the standard free groups. Indeed, in  an analog of Stallings’ foldings (see [17, 5]) was introduced for an arbitrary finitely generated subgroup of , which allows one to solve effectively the membership problem in , as well as in an arbitrary finitely generated subgroup of it. Next, in  this technique was further developed to obtain the solution of many algorithmic problems. In particular, it was proved that for a f.g. subgroups and of such that there are only finitely many conjugacy classes of intersections in . Moreover, one can find a finite set of representatives of these classes effectively. This implies that one can effectively decide whether two finitely generated subgroups of are conjugate or not, and check if a given finitely generated subgroup is malnormal in . Needless to say that all these results can be reformulated for f.g. fully residually free groups.
In the present paper we further develop these methods focusing on the problems involving finite index subgroups. It is worth mentioning that non-abelian f.g. subgroups of f.g. fully residually free groups have finite index in their normalizers - this fact follows immediately from Theorem 7  (see also ) - but no criterion for detecting finite index subgroups was known. In this paper we provide such a criterion which can be checked effectively given a finite presentation of a f.g. fully residually free group and a finite generating set of its subgroup. This allows us to draw several corollaries including an analogue of Greenberg-Stallings Theorem for free groups.
The authors are extremely grateful to Alexei G. Miasnikov for insightful discussions and many helpful comments and suggestions.
2.1 Lyndon’s free -group and infinite words
Let be a free non-abelian group with basis and be a ring of polynomials with integer coefficients in a variable . In  Lyndon introduced a -completion of , which is called now the Lyndon’s free -group.
It turns out that can be described as a union of a sequence of extensions of centralizers 
where is obtained from by extension of all cyclic centralizers in by a free abelian group of countable rank.
In  it was shown that elements of can be viewed as infinite words defined in the following way. Let be a discretely ordered abelian group. By we denote the minimal positive element of . Recall that if then the closed segment is defined as
Let be a set. An -word is a function of the type
where . The element is called the length of . By we denote the empty word. We say that is reduced if for any Then, as in a free group, one can introduce a partial multiplication , an inversion, a word reduction etc., on the set of all -words (infinite words) . We write instead of if All these definitions make it possible to develop infinite words techniques, which provide a very convenient combinatorial tool (for all the details we refer to ).
It was proved in  that can be canonically embedded into the set of reduced infinite words , where , an additive group of polynomials with integer coefficients, is viewed as an ordered abelian group with respect to the standard lexicographic order (that is, the order which compares the degrees of polynomials first, and if the degrees are equal, compares the coefficients of corresponding terms starting with the terms of highest degree). More precisely, the embedding of into was constructed by induction, that is, all from the series (1) were embedded step by step in the following way. Suppose, the embedding of into is already constructed. Then, one chooses a Lyndon’s set (see ) and the extension of cenralizers of all elements from produces , which is now also naturally embedded into .
The existence of an embedding of into the set of infinite words implies automatically the fact that all subgroups of are also subsets of , that is, their elements can be viewed as infinite words. From now on we assume the embedding to be fixed. Moreover, for simplicity we identify with its image .
2.2 Reduced forms for elements of
We may assume that the set
is well-ordered. Let
be enumeration of elements of in increasing order. Denote by the set of indices of elements from . Now has the following representation as a reduced infinite word:
where (or ) (or ), (recall that if ). Representation (2) is called -reduced if the ordered -tuple is maximal with respect to the left lexicographic order among all possible such representations of .
Suppose and . If
then there exists another representation of
The corresponding -tuples are and . In the former one maximization of exponents of goes from left to right, while in the latter one from right to left.
From (2) one can obtain another representation of . Fix any from the list . Then
where . Representation (3) is called a -representation or a -form of . In other words, to obtain a -form one has to ”mark” in (2) only nonstandard exponents of . Representation (3) is called -reduced if the ordered -tuple is maximal with respect to the left lexicographic order among all possible -forms of .
Observe that if (3) is a -form for and is cyclically reduced then obviously
 For any given -reduced form of , there exists a cyclic permutation of such that its -reduced form is cyclically -reduced.
Let have a -reduced form
where (or ), (or ), . Now, recursively one has a -reduced form for
where and one can get down to the free group with such a decomposition of , where step by step subwords between nonstandard powers of elements from are presented as -forms, . Thus, from this decomposition one can form the following series for :
where are subgroups of , which do not belong to and is obtained from by a centralizer extension of a single element . Element belongs to and does not belong to the previous terms. Series (5) is called an extension series for .
Using the extension series above we can decompose in the following way: has a -reduced form
where all in their turn are -reduced forms representing elements from . This gives one a decomposition of related to its extension series. We call this decomposition a standard decomposition or a standard representation of .
Observe that for any , its standard decomposition can be viewed as a finite product , where
We denote this product by so we have
where is a finite product in the alphabet corresponding to , and from now on, by a standard decomposition of an element we understand not the representation of as a reduced infinite word but the finite product .
By we denote a finite subset of such that if contains a letter such that then . Observe that is ordered with an order induced from , so we have
where if and . By we denote the maximal element of .
If then by we denote the maximal degree of infinite exponents of , which appear in .
It is easy to see that in general and if and only if the -reduced form of is cyclically -reduced, where .
From the definition of a Lyndon’s set and the results of  it follows that if is a Lyndon’s set then a set obtained from by cyclic decompositions of its elements is also a Lyndon’s set. Thus, by Lemma 1 we can assume a -reduced form of any to be cyclically -reduced, where . Hence, we can assume
for any .
2.3 Embedding theorems
There are three results which play an important role in this paper. The first embedding theorem is due to Kharlampovich and Myasnikov.
Theorem 1 (The first embedding theorem ())
Given a finite presentation of a finitely generated fully residually free group one can effectively construct an embedding (by specifying the images of the generators of ).
Combining Theorem 1 with the result on the representation of as a union of a sequence of extensions of centralizers one can get the following theorem.
Theorem 2 (The second embedding theorem)
Given a finite presentation of a finitely generated fully residually free group one can effectively construct a finite sequence of extension of centralizers
where is an extension of the centralizer of some element by an infinite cyclic group , and an embedding (by specifying the images of the generators of ).
Theorem 3 (The third embedding theorem)
Given a finite presentation of a finitely generated fully residually free group one can effectively construct an embedding (by specifying the images of the generators of ).
2.4 Graphs labeled by infinite -words
By an -labeled directed graph (-graph) we understand a combinatorial graph where every edge has a direction and is labeled either by a letter from or by an infinite word , denoted .
For each edge of we denote the origin of by and the terminus of by .
For each edge of -graph we can introduce a formal inverse of with the label and the endpoints defined as , that is, the direction of is reversed with respect to the direction of . For the new edges we set . The new graph, endowed with this additional structure we denote by . Usually we will abuse the notation by disregarding the difference between and .
A path in is a sequence of edges , where each is an edge of and the origin of each is the terminus of . Observe that is a word in the alphabet and we denote by a reduced infinite word (this product is always defined).
A path in is called reduced if for all .
A path in is called label reduced if
if is a subpath of such that and for any , provided , then and .
Let be a -graph and be fixed. Vertices are called -equivalent (denoted ) if there exists a path in such that and . is an equivalence relation on vertices of , so if is finite then all its vertices can be divided into a finite number of pairwise disjoint equivalence classes. Suppose, is fixed. One can take the subgraph of spanned by all the vertices which are -equivalent to and remove from it all edges with labels not equal to . We denote the resulting subgraph of by and call a -component of . If then one can define a set
 Let be a -graph and . Then
is a subgroup of ;
is isomorphic to a subgroup of ;
if is a finite graph, then is finitely generated;
if then .
Following  one can introduce operations on -components which are called -foldings. One of the most important properties of -foldings is that they do not change subgroups associated with -components.
 Let be a -graph, and be finite. Then there exist a -graph obtained from by finitely many -foldings such that corresponds to and consists of a simple positively oriented path , and some edges that are not in connecting some pairs of vertices in .
in Lemma 3 is called a reduced -component. Since is a simple path there exists a vertex which is an origin of only one positive edge in . is called a base-point of .
It turns out that any finite reduced -component in a -graph is characterized completely by the pair in the following sense. For any reduced path in there exists a unique reduced subpath (denoted ) of with the same endpoints as , such that . Moreover, let , where and let be reduced subpaths of such that . The set of paths is called a set of path representatives associated with (denoted by ).
 Let be a finite reduced -component in a -graph , and let . If for any then either there exists a unique reduced path in such that and or there exists no path in with this property.
If is reduced and satisfies the condition from Lemma 4 then we call a -folded -component.
2.5 Languages associated with -graphs
Let be a -graph and let be a vertex of . We define the language of with respect to as
 Let be a finite -graph and let . Then is a subgroup of .
Let be a -graph and be a reduced path in . Let and let
be the standard decomposition of , where . We write
if can be subdivided into subpaths
where each is a path in some -component of so that , and each is a path in which does not contain edges labeled by so that the equality is defined inductively in the same way. Observe that if then if and only if and .
Let be a finite -graph. Since is finite, the set of elements such that there exists an edge in labeled by is finite and ordered with the order induced from . Thus one can associate with an ordered set for .
Let be fixed and be a subgraph of which consists only of edges such that either or . is called an -level graph of (by -level graph we understand a subgraph of which consists only of edges with labels from ) and the level (denoted ) of is the minimal such that . Observe that may not be connected for some , but still one can apply to partial and -foldings, .
A finite connected -graph is called -folded if for any reduced path in with there exists a unique label reduced path such that .
The above definition is equivalent to a more technical one given in .
 Let be a finite connected -graph and . Then there exists a -folded -graph and such that . Moreover can be constructed effectively by adding to finitely many edges and applying finitely many free and -foldings.
 Let be a finitely generated subgroup of . Then there exists a -folded -graph and a vertex of such that .
 There is an algorithm which, given finitely many standard decompositions of elements from , constructs a -folded -graph , such that .
The properties of -folded graphs make it possible to solve the membership problem in finitely generated subgroups of .
 Every finitely generated subgroup of has a solvable membership problem. That is, there exists an algorithm which, given finitely many standard decompositions of elements from , decides whether or not belongs to the subgroup of .
3 Finite index criteria
It is not difficult to check if a finitely generated subgroup of a free group is of finite index (see, for example, ). This can be done by checking if the normal form of every element of is “readable” in a folded graph corresponding to . Similar result can be easily proved for finitely generated subgroups of .
Let be a finitely generated subgroup of and . Then the following are equivalent:
there exists a finite -folded ()-graph with a vertex such that and for every there exists a path in such that .
Proof. At first, assume . Hence, there exist such that
Take a finite -folded ()-graph with a vertex such that . For each take a path labeled by and glue its initial end-point to at . The resulting ()-graph by Proposition 1 can be transformed into a -folded ()-graph whose language is . But since for every product there exists a path in such that , this property also holds in .
Now, assume that there exists a finite -folded ()-graph with a vertex such that and for every there exists a path in such that . For every there exists a path such that . Since is finite, the set of such paths is finite and their reduced labels obviously can be taken to be representatives of right cosets in by .
At the same time, it is important to understand that not every -folded graph representing a subgroup of finite index in has the property that the normal form of every element of is “readable” in it.
Let and , where , Without loss of generality we can assume to be such that the graphs shown on Figure 1 are -folded. Observe that , but is not “readable” in the graph defining .
Hence, “readability” of normal forms of elements from is a too strong property to work with. Instead, below we develop the idea of “readability” of infinite paths arising in a -folded graph for , and show that every such infinite path is readable in any -folded graph for as long as .
3.1 Equivalence of infinite powers
For any , denote
Let be a finite subset of . Denote . For any we say that is -equivalent to , where , and denote , if the following condition holds: for any if is a standard form, then is a standard form as well, provided does not end with a power of and does not start with a power of .
For every finite and splits into a finite number of -equivalence classes. Moreover, for any , the equivalence class of can be effectively constructed.
Proof. Fix a finite set and .
For a given occurrence , describe all that can be replaced with, not breaking the standard form. Suppose some does break the standard form. Enumerate possible reasons according to definition of standard form:
is no longer reduced, which means that is of opposite sign to ,
, and either ends with , or starts with ,
, where either or , and either ends with , or starts with ,
, where either or , and either ends with , or starts with .
As we can see, all possible cases result in conditions of the following types
Since there are only finitely many occurrences of in standard forms of , these condition split into finitely many classes and each class can be easily constructed.
For every finite and , -equivalence classes in can be effectively described and enumerated.
Proof. Follows immediately from the previous result.
3.2 Standard form types associated to ()-graph
Let be a finite -folded ()-graph. We fix for the rest of this subsection. Observe that is a finite subset of and for each , by Proposition 6, there are only finitely many -equivalence classes in .
Fix . Let be a -component of . By Lemma 2, is a subgroup of . For every define the set of reduced paths in from to and a subset of as follows
Note that for any reduced path in from to . That is, is a (right) coset in by , and it is completely defined by a finite number of generators of and a representative .
For a -component of , a pair , and a -equivalence class in , consider a subset of defined as
Paths and which belong to a certain we call -equivalent.
For every , there are only finitely many sets of the type . Moreover, the set of labels of paths from each can be effectively described and enumerated.
Proof. The first part of the statement follows from the fact that the number of tuples is finite (in particular, by Proposition 6).
Next, by definition, the set of labels of paths from is , where both sets can be effectively described and enumerated (in particular, by Corollary 1). Finally, since these sets are subsets of an abelian group , the intersection also can be effectively described and enumerated.
Denote by the set of all paths in and define the set of special paths as follows
For a vertex , similarly define
Observe that for every