Local similarity groups with context-free co-word problem
Let be a group, and let be a finite subset of that generates as a monoid. The co-word problem is the collection of words in the free monoid that represent non-trivial elements of .
A current conjecture, based originally on a conjecture of Lehnert and modified into its current form by Bleak, Matucci, and Neuhöffer, says that Thompson’s group is a universal group with context-free co-word problem. In other words, it is conjectured that a group has a context-free co-word problem exactly if it is a finitely generated subgroup of .
Hughes introduced the class of groups that are determined by finite similarity structures. An group acts by local similarities on a compact ultrametric space. Thompson’s group is a representative example, but there are many others.
We show that groups have context-free co-word problem under a minimal additional hypothesis. As a result, we can specify a subfamily of groups that are potential counterexamples to the conjecture.
Key words and phrases:Thompson’s groups, context-free languages, push-down automata
2000 Mathematics Subject Classification:20F10, 03D40
Let be a group, and let be a finite subset that generates as a monoid. The word problem of with respect to , denoted , is the collection of all positive words in such that represents the identity in ; the co-word problem of with respect to , denoted , is the set of all positive words that represent non-trivial elements of . In this point of view, both the word and the co-word problem of are formal languages, which suggests the question of placing these problems within the Chomsky hierarchy of languages.
Anisimov [Anisimov] proved that is a regular language if and only if is finite. A celebrated theorem of Muller and Schupp [MullerSchupp] says that a finitely generated group has context-free word problem if and only if it is virtually free. (A language is context-free if it is recognized by a pushdown automaton.) In this case, as noted in [Holt], the word problem is actually a deterministic context-free language. Shapiro [Shapiro] described sufficient conditions for a group to have a context-sensitive word problem.
Since the classes of regular, deterministic context-free, and context-sensitive languages are all closed under taking complements, it is of no additional interest to study groups with regular, deterministic context-free, or context-sensitive co-word problems, since the classes of groups in question do not change. The (non-deterministic) context-free languages are not closed under taking complements, however, so groups with context-free co-word problem are not (a priori, at least) the same as groups with context-free word problem.
Holt, Rees, Röver, and Thomas [Holt] introduced the class of groups with context-free co-word problem, denoted . They proved that all finitely generated virtually free groups are , and that the class of groups is closed under taking finite direct products, passage to finitely generated subgroups, passage to finite index overgroups, and taking restricted wreath products with virtually free top group. They proved negative results as well: for instance, the Baumslag-Solitar groups are not if , and polycyclic groups are not unless they are virtually abelian. They conjectured that groups are not closed under the operation of taking free products, and indeed specifically conjectured that is not a group.
Lehnert and Schweitzer [Lehnert] later showed that the Thompson group is . Since seems to contain many types of subgroups (among them all finite groups, all countable free groups, and all countably generated free abelian groups), this raised the possibility of showing that is by embedding the latter group into . Bleak and Salazar-Díaz [Bleak1], motivated at least in part by these considerations, proved that does not embed in (leaving the conjecture from [Holt] open), and also established the existence of many embeddings into . The basic effect of their embedding theorems is to show that the class of finitely generated subgroups of Thompson’s group is closed under the same operations as those from [Holt], as listed above.
The similarity between the classes and seems to have led to the following conjecture:
The classes and are the same; i.e., Thompson’s group is a universal group.
Lehnert had conjectured in his thesis that a certain closely related group of quasi-automorphisms of the infinite binary tree is a universal group. Bleak, Matucci, and Neuhöffer [Bleak2] established the existence of embeddings from to and from to . As a result, Lehnert’s conjecture is equivalent Conjecture 1.1. We refer the reader to the excellent introductions of [Bleak2] and [Bleak1] for a more extensive discussion of these and related questions.
Here we show that many groups defined by finite similarity structures are contained in . The precise statement is as follows.
Let be a compact ultrametric space endowed with a finite similarity structure . Assume that there are only finitely many -classes of balls.
For any finitely generated subgroup of and finite subset of that generates as a monoid, the co-word problem is a context-free language.
The groups defined by finite similarity structures (or groups) were first studied by Hughes [Hughes1], who showed that all groups act properly on CAT(0) cubical complexes and (therefore) have the Haagerup property. Farley and Hughes [FarleyHughes1] proved that a class of groups have type . All of the latter groups satisfy the hypotheses of the main theorem, so all are also groups. (We note that the main theorem also covers as a special case.)
The class of groups is not well-understood, but we can specify a certain subclass that shows promise as a source of counterexamples to Conjecture 1.1. These are the Nekrashevych-Röver examples from [FarleyHughes1] and [Hughes1]. The results of [FarleyHughes1] show that most of these examples are not isomorphic to (nor to the -ary versions of ), and it is not difficult to show that they do not contain as a subgroup of finite index. It seems to be unknown whether there are any embeddings of these groups into . Our main theorem therefore leaves Conjecture 1.1 open.
(Note that the Nekrashevych-Röver examples considered in [FarleyHughes1] and [Hughes1] are not as general as the classes of groups from [Rov99] and [NekJOT]; the finiteness of the similarity structures proves to be a somewhat restrictive hypothesis.)
The proof of the main theorem closely follows the work of Lehnert and Schweitzer [Lehnert]. We identify two main ingredients of their proof:
All of the groups satisfying the hypothesis of the main theorem admit test partitions (Definition 3.1). That is, there is a finite partition of the compact ultrametric space into balls, such that every non-trivial word in the generators of has a cyclic shift that moves at least one of the balls off of itself, and
for each pair of distinct balls , where and are from the test partition, there is a “-witness automaton”, which is a pushdown automaton that can witness an element moving part of into .
The main theorem follows very easily from (1) and (2). The proofs that (1) and (2) hold are complicated somewhat by the generality of our assumptions, but are already implicit in [Lehnert]. Most of the work goes into building the witness automata. We describe a stack language that the witness automata use to describe, store, and manipulate metric balls in . One slight novelty (not present or necessary in [Lehnert]) is that the witness automata write functions from the similarity structure on their stacks and make partial computations using these functions.
We briefly describe the structure of the paper. Section 2 contains a summary of the relevant background, including string rewriting systems, pushdown automata, groups, and standing assumptions. Section 3 contains a proof that the groups admit test partitions, as described above. Section 4 describes the stack language for the witness automata, and Section 5 gives the construction of the witness automata. Section 6 collects the ingredients of the previous sections into a proof of the main theorem.
2.1. String Rewriting Systems
A rewrite system is a directed graph . We write if and are vertices of and there is a directed edge from to . We write if there is a directed path from to . The rewrite system is called locally confluent if whenever and , there is some such that and . The rewrite system is confluent if whenever and , there is some such that and . The rewrite system is terminating if there is no infinite directed path in . If a rewrite system is both terminating and confluent, then we say that it is complete. A vertex of is called reduced if it is not the initial vertex of any directed edge in .
[Newman] Every terminating, locally confluent rewrite system is complete. ∎
The relation generates an equivalence relation on the vertices of . It is not difficult to see that each equivalence class in this equivalence relation contains a unique reduced element in the event that is complete.
Let be a finite set, called an alphabet. Let be a subset of the free monoid . Let be a collection of relations (or rewriting rules) of the form , where , . (Thus, the are positive words in the alphabet , either of which may be empty. The are not required to be in .)
We define a string rewriting system as follows: The vertices are words from . For , , there is a directed edge whenever there are words , such that and , for some .
2.2. Pushdown Automata
Let and be finite sets. The set is the input alphabet and is the stack alphabet. The stack alphabet contains a special symbol, , called the initial stack symbol.
A (generalized) pushdown automaton (or PDA) over and is a finite labelled directed graph endowed with an initial state and a (possibly empty) collection of terminal states . Each directed edge is labelled by a triple , where denotes an empty string.
Each PDA accepts languages either by terminal state, or by empty stack, and this information must be specified as part of the automaton’s definition. See Definition 2.7.
Let be a pushdown automaton. We describe a class of directed paths in , called the valid paths, by induction on length. The path of length starting at the initial vertex is valid; its stack value is . Let () be a valid path in , where is the edge that is crossed first. Let be an edge whose initial vertex is the terminal vertex of ; we suppose that the label of is . The path is also valid, provided that the stack value of has as a prefix; that is, if the stack value of has the form . The stack value of is then . We let denote the stack value of a valid path .
The label of a valid path is , where is the first coordinate of the label for (an element of , or the empty string). The label of a valid path will be denoted .
Let be a PDA. The language accepted by is either
, if accepts by empty stack, or
, if accepts by terminal state.
A subset of the free monoid is called a (non-deterministic) context-free language if it is , for some pushdown automaton .
The class of languages that are accepted by empty stack (in the above sense) is the same as the class of languages that are accepted by terminal state. That is, given an automaton that accepts a language by empty stack, there is another automaton that accepts by terminal state (and conversely).
All of the automata considered in this paper will accept by empty stack.
The functioning of an automaton can be described in plain language as follows. We begin with a word written on an input tape, and the word written on the memory tape (or stack). We imagine the stack as a sequence of boxes extending indefinitely to our left, all empty except for the rightmost one, which has written in it. Our automaton reads the input tape from right to left. It can read and write on the stack only from the left (i.e., from the leftmost nonempty box). Beginning in the initial state , it can follow any directed edge it chooses, provided that it meets the proper prerequisites: if the label of is , then must be the rightmost remaining symbol on the input tape, and the word must be a prefix of the word written on the stack. If these conditions are met, then it can cross the edge into the next state, simultaneously erasing the letter from the input tape, erasing from the left end of the stack, and then writing on the left end of the stack. The original input word is accepted if the automaton can reach a state with nothing left on its input tape, and nothing on its stack (not even the symbol ).
We note that a label such as describes an empty set of prerequisites. Such an arrow may always be crossed, without reading the input tape or the stack, no matter whether one or the other is empty.
2.3. Review of ultrametric spaces and finite similarity structures
We now give a quick review (without proofs) of finite similarity structures on compact ultrametric spaces, as defined in [Hughes1]. Most of this subsection is taken from [FarleyHughes1].
An ultrametric space is a metric space such that
for all .
Let be an ultrametric space.
Let be an open metric ball in . If , then .
If and are open metric balls in , then either the balls are disjoint, or one is contained in the other.
If is compact, then each open ball is contained in at most finitely many distinct open balls of , and these form an increasing sequence:
If is compact and is not an isolated point, then each open ball is partitioned by its maximal proper open subballs, which are finite in number.
Throughout this paper, “ball” will always mean “open ball”.
Let be a function between metric spaces. We say that is a similarity if there is a constant such that , for all and in .
A finite similarity structure for is a function that assigns to each ordered pair of balls in a (possibly empty) set of surjective similarities such that whenever are balls in , the following properties hold:
(Finiteness) is a finite set.
(Inverses) If , then .
(Compositions) If and , then
(Restrictions) If and , then
A homeomorphism is locally determined by provided that for every , there exists a ball in such that , is a ball in , and .
The finite similarity structure (FSS) group is the set of all homeomorphisms such that is locally determined by .
The fact that is a group under composition is due to Hughes [Hughes1].
([Hughes1], Definition 3.6) If , then we can choose a partition of by balls such that, for each , is a ball and . Each element of this partition is called a region for .
2.4. Standing Assumptions
In this section, we set conventions that hold for the rest of the paper.
We say that two balls and are in the same -class if the set is non-empty.
We assume that is a compact ultrametric space with finite similarity structure . We assume that there are only finitely many -classes of balls, represented by
We let denote the -class of a ball , and let .
Each ball is related to exactly one of the . We choose (and fix) an element . We choose .
Each ball has a finite collection of maximal proper subballs, denoted
This numbering (of the balls and their maximal proper subballs) is fixed throughout the rest of the argument. We let .
We will let denote a finitely generated subgroup of (see Definition 2.17). We choose a finite set that generates as a monoid, i.e., each element can be expressed in the form , where , , and only positive powers of the are used. We choose (and fix) regions for each .
3. Test Partitions
Let be a finite partition of . We say that is a test partition if, for any word in the generators , whenever
for all and , then .
If is a compact ultrametric space and , then is a finite partition of by open balls.
This follows easily from Lemma 2.12(1). ∎
Let be chosen so that
We let . This is the big partition.
Note that, for each and , is contained in a unique region of .
Let be a compact ultrametric space, and let be a finite group of isometries of . There is an such that if acts trivially on , then .
For each nontrivial , there is such that . We choose satisfying
We set . Now suppose that and acts trivially on . Thus , so , but
a contradiction. ∎
Write . For each (), we can choose to meet the conditions satisfied by in the previous lemma, for . Let . Let . This is the small partition.
The small partition is a test partition.
Let be a word in the generators ; we assume . We suppose, for a contradiction, that for all ,
for all . Since , we can find such that .
Fix . For each , there is an open ball , with , such that:
lies in a region of , for and
for at least one .
We first prove that, for any , there is a ball neighborhood of satisfying (1). We choose and fix .
Consider the elements . We first observe that there is a constant such that if any ball lies inside a region for (for any ), then stretches by a factor of no more than . Next, observe that there is a constant such that any ball of diameter less than or equal to lies inside of a region for , for all . It follows easily that any ball of diameter less than satisfies (1); we can clearly choose some such ball, to be a neighborhood of . We note that if a ball satisfies (1), then so does every subball.
be the collection of all balls containing . (Thus, each is a maximal proper subball inside , for .) There is a maximal ball , , such that satisfies (1).
If , then the entire composition . We then take such that . The required is .
Now assume that . There is some such that is a ball and
but properly contains a region for ; let be the regions of that are contained in . We must have for some (by maximality of in ); the reverse containment follows, since is contained in a region for by our assumptions.
Now note that is partitioned by elements of ; there is some such that . We have that the map is a map from the similarity structure. The required ball is . ∎
Apply the sub lemma to : there is (an open ball) with the given properties. Let be such that
Since is invariant under every cyclic permutation of by our assumption,
Our assumptions imply that and are both in , and both are bijections.
Consider . It must be that ; if , then
which implies that , a contradiction.
Now, since , it moves some element of . ∎
4. A language for
In this section, we introduce languages and . The language will serve as the stack language for the witness automata of Section 5. The language consists of the reduced elements of ; it is useful because there is a one-to-one correspondence between elements of and metric balls in .
4.1. The languages and
We define a language as follows. The alphabet for consists of the symbols:
, the initial stack symbol;
, , .
(We refer the reader to Convention 2.21 for the meanings of and .) The language consists of all words of the form
where and for . (Here, and in what follows, we make the convention that ; i.e., that .)
The language also uses symbols of the form , where . The general element of takes the form
where each () is a word in the symbols , and some or all of the might be empty.
The letter signifies a ball of similarity class ; the signifies that it is the th maximal proper subball of the ball before it in the sequence. The letter signifies the top ball, . Refer to Convention 2.21 for the significance of and .
The condition for is designed to insure that each ball has the correct type; i.e., that the sequence encodes consistent information about the similarity types of subballs.
Let denote the collection of all metric balls in . We define an evaluation map by sending
where if .
Let . We say that is a prefix of if with the initial stack symbol omitted is a prefix of in the usual sense; that is , for some string .
The function is a bijection.
Moreover, a word is a proper prefix of if and only if is a proper subball of , and is a maximal proper prefix of if and only if is a maximal proper subball of .
We first prove surjectivity. Let be a ball in . We let
be the collection of all balls in that contain . (Thus, is a maximal proper subball in for .)
In the diagram
the balls and the maps are the ones given in Convention 2.21; the unlabeled arrows are inclusions. Note, in particular, that the maps are bijections taken from the -structure, and that is the identity map. If we follow the arrows from to , the corresponding composition is a member of that carries the ball to a maximal proper subball of . Supposing that the number of the latter maximal proper subball is (see Convention 2.21), we obtain a diagram
where , for . This diagram commutes “up to images”: that is, if we start at a given node in the diagram, then the image of that first node in any other node is independent of path. (The diagram is not guaranteed to commute in the usual sense.) Set . We note that
where the first equality is the definition of , the second follows since , the third follows from the commutativity of the diagram up to images, and the fourth follows from surjectivity of . This proves that is surjective.
Before proving injectivity of , we note that, for a given
each of the functions maps its domain onto a proper subball of its codomain. As a result, a word is a proper prefix of if and only if is a proper subball of , and is a maximal proper prefix of if and only if is a maximal proper subball of .
Suppose now that , for some , . By the above discussion, we can assume that neither nor is a prefix of the other. Let be the largest common prefix of and . Let . Since and for non-trivial strings and with different initial symbols, and are disjoint proper subballs of . ∎
Let be a ball in . The address of is the inverse image of under the evaluation map , but with the initial stack symbol omitted. We write .
4.2. A string rewriting system based on
In this subsection, we describe a string rewriting system with underlying vertex set . The witness automata of Section 5 will use this rewrite system to perform partial calculations in on their stacks.
by the rule , for . (We recall that is the number of -classes of balls in ; see Convention 2.21.) The union is over all pairs of balls .
If , where for some , then is the standard representative of , and is the standard form for .
If for some , then we sometimes confuse with itself; this is justified by our choices in Convention 2.21.
Define a string rewriting system as follows. The vertices are elements of the language . There are four families of rewriting rules:
where is a standard form; i.e., ;
where , for some ;
for arbitrary ;
We note that the total number of the above rules is finite, since there are only finitely many -classes of balls.
The string rewriting system is locally confluent and terminating. Each reduced element of is in . The function is constant on equivalence classes modulo .
It is clear that each reduced element of is in , and that is terminating. Local confluence of is clear, except for one case, which we will now consider.
Suppose that contains a substring of the form . We can apply two different overlapping rewrite rules to , one sending to , and the other sending to . We need to show that and flow to a common string. Note that
It therefore suffices to demonstrate that the maps and are equal. But this follows from the commutativity of the following diagram:
where the vertical arrows are the canonical identifications from Convention 2.21 (e.g., the first vertical arrow is ). It now follows that is locally confluent and terminating.
We now prove that is constant on the equivalence classes modulo . It is clear that applications of rules (2)-(4) do not change the value of ; we check that (1) also does not change the value of . Suppose we are given
where each , and is therefore , for in the form given in Definition 4.1. We pick a particular