Efficient solution of some problems in free partially commutative monoids

Divisibility theory and complexity of algorithms for free partially commutative groups

“…Wrathal [62] found an efficient algorithm for solving the conjugacy problem in G by reducing it to the conjugacy problem in the free partially commutative monoid M = M(X ∪ X −1 ), which, in its turn, is solved in [48] by a reduction to pattern-matching questions (recall that two elements u, v ∈ M are conjugate if their exists z ∈ M such that uz = zv). In this section we shall give a direct solution to the conjugacy problem for partially commutative groups in terms of a conjugacy criterion for HNN-extensions and the divisibility theory developed in the preceding section.…”

Compressed Conjugacy and the Word Problem for Outer Automorphism Groups of Graph Groups

“…The equivalence of (1) and (2) can be found in [29], the equivalence of (2) and (3) is shown in [16]. We can now infer Theorem 1:…”

“…For this, we will follow the approach from [16,29] for non-compressed traces. The following result allows us to transfer the conjugacy problem to a problem on (compressed) traces:…”

“…For our polynomial time algorithm, we have to develop a pattern matching algorithm for SLP-compressed Mazurkiewicz traces, which is inspired by a pattern matching algorithm for hierarchical message sequence charts from [12]. For the noncompressed version of the conjugacy problem in G(Σ, I), a linear time algorithm was presented in [29] based on [16]. In [7] this result was generalized to various subgroups of graph groups.…”

Cryptography with right-angled Artin groups

“…In the mentioned paper, for example, the authors apply their cryptosystem for small cancellation groups. In the case of right-angled Artin groups, the easiness of the word problem was first proved in a paper by Liu-Wrathall-Zeger [20] which in a more general framework of free partially commutative monoids, describes an algorithm which is effective in linear polynomial time. More recently, Crisp-Goddelle-Wiest [21] have extended this result (with different methods) to some families of subgroups of right-angled Artin groups, as for example braid groups.…”