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«The conjugacy problem in subgroups of right-angled Artin groups John Crisp, Eddy Godelle, and Bert Wiest Abstract We prove that the conjugacy problem ...»

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Submitted exclusively to the London Mathematical Society

doi:10.1112/0000/000000

The conjugacy problem in subgroups of right-angled Artin groups

John Crisp, Eddy Godelle, and Bert Wiest

Abstract

We prove that the conjugacy problem in a large and natural class of subgroups of right-angled

Artin groups (RAAGs), can be solved in linear-time. This class of subgroups has been previously

studied by Crisp and Wiest, and independently by Haglund and Wise, as fundamental groups

of compact special cube complexes.

1. Introduction It is well known that the conjugacy problem in free groups can be solved in linear-time by a RAM (random access memory) machine. This result has been generalized in two different directions. On the one hand, Epstein and Holt [14] have shown that the conjugacy problem is linear in all word-hyperbolic groups. On the other hand, Liu, Wrathall and Zeger have proved the analogue result for all right-angled Artin groups ([24], based on [31]). Note that the latter groups are also called “partially commutative groups” or “graph groups” in the literature.

The aim of the present paper is to further generalize the second result, by proving linearity of the conjugacy problem in a large class of subgroups of right-angled Artin groups. This class of groups has previously been studied by Crisp and Wiest [11, 12], and independently by Haglund and Wise [19]; these groups are fundamental groups of certain cube complexes, called compact A-special cube complexes in [19].

Our generalization is based on a new solution to the conjugacy problem in right-angled Artin groups (RAAGs), different from the one of Liu, Wrathall and Zeger, but rather close in spirit to the methods of Lalonde and Viennot [23, 30].

The class of groups considered in this paper contains in particular all graph braid groups [1, 2, 15, 16, 26] and more generally all state complex groups [3, 17], which are closely related to robotics [2, 17] – indeed, our results can be interpreted as giving very efficient algorithms for motion planning of periodic robot movements. Notice that it is still unknown whether Coxeterand Artin groups are virtually fundamental groups of compact special cube complexes [19, 20], so our results do not apply immediately to these groups.

The plan of the paper is as follows. In the second section we present our new solution to the conjugacy problem in RAAGs. In the third section we give a precise definition of the class of subgroups of RAAGs under consideration, and prove that they inherit a linear-time solution to the conjugacy problem from their supergroups.

2. The conjugacy problem in RAAGs is linear-time We recall that a right-angled Artin group is a group given by a finite presentation, where every relation states that some pair of generators commutes. Graphically, a right-angled Artin group A can be specified by a simple graph ΓA, where the generators of A correspond to the vertices of ΓA, and a pair of generators commutes if and only if the corresponding vertices are 2000 Mathematics Subject Classification 00000.

Page 2 of 19 JOHN CRISP, EDDY GODELLE, AND BERT WIEST not connected by an edge. Note that the opposite convention (connecting commuting generators by an edge) is also very common, but in the present paper we shall stick to this convention.

Right-angled Artin groups have been widely studied in the last decades – see [10] for an excellent survey. Several solutions to the word and conjugacy problem have been found. It seems difficult to have a complete bibliography concering these two problems, but the articles by Servatius [27], Van Wyk [29], Liu, Wrathall and Zeger [24] (based on [31]), Cartier and Foata [9], Viennot [30], Lalonde [23], and Krob, Mairesse, and Michos [22] can be highlighted.

In order to approach the conjugacy problem in right-angled Artin groups, let us first consider the relatively easy case of free groups. Given two cyclic words of length ℓ1 and ℓ2 respectively,

there is a two step algorithm which can be performed in time O(ℓ1 + ℓ2 ) on a RAM machine:

first each word can be cyclically reduced in time O(ℓ1 ) and O(ℓ2 ), respectively. If the reduced words have different lengths, then they are not conjugate. If they have the same length ℓ, then they can be compared in time O(ℓ) using standard pattern matching algorithms, like the Knuth-Morris-Pratt algorithm, the Boyer-Moore algorithm, or algorithms based on suffixtree methods – see [21, 7, 4, 18, 28]. It should be stressed that on a Turing machine these algorithms take time O(ℓ log(ℓ)).

In the sequel, we assume that A is a fixed right-angled Artin group given by a fixed presentation. We denote by {a1, · · ·, aN } the generating set of A associated with this presentation.

In this section we shall provide an algorithm for the conjugacy problem in A which works as follows: given a word w, another word w′ with smaller or equal length is created in linear time such that w and w′ represent conjugate elements of A. Furthermore, the word w′ depends only on the conjugacy class in A of the element represented by w, up to a cyclic permutation of its letters. This yields a linear-time solution to the conjugacy problem in A because, given words w and v we can compute the canonical cyclic words w′ and v ′ representing their conjugacy classes, and compare those by one of the algorithms mentioned above.





2.1. Normal forms and pilings In this subsection we reprove the well-known fact that there is a linear-time solution to the word problem. We start by recalling the following classical lemma.

Lemma 2.1.

[27] Any element of A can be represented by a reduced word (one which does not contain a subword of the form a±1 xa∓1, where all letters of x commute with ai ).

i i Moreover, any two reduced representatives of the same element are related by a finite number of commutation relations – no insertions/deletions of trivial pairs are needed.

Now we introduce our main tool, the notion of a piling.

Definition 2.2. An

Abstract

piling is a collection of N words, one for each generator ai of A, over the alphabet with three symbols {+, −, 0}.

The word associated with the generator ai will be called the ai -stack of the abstract piling.

The product of two abstract pilings is defined as the piling obtained by concatenation of the corresponding stacks.

–  –  –

is. If this letter is different from −ǫ (the no-cancellation cases) then we append a letter + or − (the sign of ǫ) at the end of the ai -stack of the piling. Moreover, we also append a letter 0 at the end of each of the aj -stacks associated with a generator aj that does not commute with the generator ai. On the other hand, if the last letter of the ai -stack is −ǫ (the cancellation case), then we erase this last letter, and we also erase the terminal letter of each of the aj -stacks of the piling associated with a generator aj that does not commute with the generator ai – note that the terminal letter of the aj -stack is necessarily “0”.

Definition 2.3. A piling is an abstract piling in the image of the function π ⋆. The set of pilings is denoted Π.

We observe that the number of letters + and − occuring in the piling π ⋆ (w) is at most equal to the length of the word w. Moreover, it is immediate from the description of the function π ⋆ that, given a word w of length ℓ, the piling π ⋆ (w) can be calculated in time O(ℓ) (linear-time).

It may be helpful to keep in mind the following physical interpretation of a piling: we have N vertical sticks, labelled by the generators a1,..., an, with beads on it; the beads are labelled by +, − or 0 such that when reading from bottom to top the sequence of labels of the beads on the ai -stick, we obtain the ai -stack of the piling. A letter ai or a−1 of the word w corresponds to i a set of beads (which we call a tile), consisting of one bead labelled + or − on the corresponding stick, and one bead labelled 0 on each of the sticks corresponding to generators of A which do not commute with ai ; each 0 labelled bead is connected to the ± labelled bead by a thread.

On a stick, adjacent 0-beads can commute with (“slide through”) each other, but 0-beads do not commute with ±-beads.

–  –  –

The map π ⋆ induces a well-defined function π : A → Π because words representing the same element of A have the same image under π ⋆ : the image of a word is unchanged by applying a commutation relation, and by inserting or deleting a trivial pair ai a−1 or a−1 ai. Now, i i from the definitions it is immediate that no cancellation occurs during the construction of the piling π ⋆ (w) of a reduced word w. Thus the identity of A is the unique element of A whose image by π is the trivial piling, and therefore the word problem is solved in linear-time: a word w represents the identity if and only if its piling π ⋆ (w) is trivial; this piling can be built in linear-time.

Page 4 of 19 JOHN CRISP, EDDY GODELLE, AND BERT WIEST

–  –  –

We remark that all the factors of a normal word are normal words.

Proposition 2.6. Any element of A has a unique normal reduced representative word.

–  –  –

In the sequel, we call this unique normal reduced word representing a the normal form of a.

There is a linear-time algorithm that associates to each piling p Proposition 2.7.

a normal word σ ⋆ (p) such that π ⋆ (σ ⋆ (p)) = p. Furthermore, for any element a of A the word σ ⋆ (π(a)) is the normal form of a.

–  –  –

representative of a. By induction, the whole word σ ⋆ (p) is a reduced representative of a.

Moreover, the word σ ⋆ (p) is initially normal, by construction, and by induction its suffix σ ⋆ (p1 ) is normal. Hence the whole word σ ⋆ (p) is normal.

Example 2.8.

Using the notation of Example 2.4, the word σ ⋆ (p) is equal to a−1 a3 a−1 a1 a2 a−1 a2 a2.

The calculation is shown in Figure 2.

–  –  –

2.2. Cyclic normal forms and pyramidal pilings We are now ready to attack the conjugacy problem.

2.2.1. Cyclically reduced words and cyclically reduced pilings. We recall that a cycling of a reduced word w is the operation of removing the first letter of the word, and placing it at the end of the word. A word is called cyclically reduced if it is reduced and if any word obtained from it by a sequence of cyclings and commutations is still reduced – in other words, if it is not of the form x1 a±1 x2 a∓1 x3, where all the letters of x1 and x3 commute with ai. As far as i i we know, all known solutions to the conjugacy problem in RAAGs are based on the following lemma.

Lemma 2.9.

Two cyclically reduced words represent conjugate elements if and only if they are related by a sequence of cyclings and commutation relations.

–  –  –

Definition 2.10. If, in a piling p, the ai -stack starts (resp. finishes) with a letter + or −, the bottom ai -tile (resp. the top ai -tile) of p is the sub-piling formed by the first (resp. last) letter of the ai -stack and the first (resp. last) letter of the aj -stacks such that ai and aj do not commute in A.

Example 2.11.

With the notation of Example 2.4, Figure 3 gives an example of top and bottom tiles of a piling.

–  –  –

Figure 3. A top a2 -tile and a bottom a2 -tile, and the associated cyclic reduction Definition 2.

12. If in a piling p the ai -stack starts with the letter + and ends with −, or vice versa, a cyclic reduction is the act of removing both top and bottom ai -tiles. We say that the piling is cyclically reduced if no cyclic reduction is possible.

Note that cyclically reducing a piling yields again a piling. We remark that there is an obvious linear-time algorithm for transforming any piling into a cyclically reduced one by a finite sequence of cyclic reductions. We also observe that for a reduced word w ∈ {a±1,..., a±1 }∗, 1 N cycling of w corresponds to a cycling of its piling, and that w is cyclically reduced if and only if the piling π ⋆ (w) is.

Now we have a fast algorithm for cyclically reducing words and pilings. In contrast to the case of free groups, however, the reduced words which can obtained are not unique up to cyclic permutation. In order to circumvent this problem, we shall introduce in the sequel the notion of a cyclic normal form.

2.2.2. Non-split words and non-split pilings. Our first objective is to restrict the conjugacy problem to the case of non-split cyclically reduced words (or pilings). We recall that a graph ΓA is associated to the right-angled Artin group A.

–  –  –

Now, if w is a cyclically reduced word that is split, then it is equivalent to a product w1 · · · wk of non-split cyclically reduced words, one for each connected component ∆i (w) of the graph ∆(w); the graph ∆i (w) is equal to ∆(wi ). Furthermore, once that the connected components ∆1 (w),..., ∆k (w) of ∆(w) are computed, appropriate words w1,..., wk can be obtained in linear-time.

Remark 2.14.

The following observation will be crucial: if v is another cyclically reduced

word, then w and v represent conjugate elements if and only if two conditions are satisfied:

firstly the graph ∆(v) is equal to ∆(w); secondly, if v1,..., vk are words such that ∆(vi ) = ∆i (v) and such that v is equivalent to the product v1 · · · vk, then for each index i the words wi and vi represent conjugate elements.

Therefore, in order to obtain a solution to the conjugacy problem in linear-time it is enough to consider the case of cyclically reduced non-split words.

2.2.3. Pyramidal piling and cyclic normal form To solve the conjugacy problem, we associate in the sequel a cyclic normal word to each cyclically reduced non-split word. We first do the analogue of this in the framework of pilings: to each non-split cyclically reduced piling, we associate a pyramidal piling.



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