Free Partially Commutative Groups

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.…”

“…Paper [10] discusses applications to geometry and topology, while major statistical characteristics for these groups are estimated in [20,58,59] and the Poisson-Furstenberg boundary described in [50]. Some algorithmic problems for right angled groups were solved in [38,62].…”

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.…”