Using Rewriting Techniques to Solve the Generalized Word Problem in Polycyclic Groups
In this paper we apply rewriting techniques to the generalized word problem GWP in polycyclic groups. We assume the group <italic>G</italic> to be given by a canonical polycyclic string-rewriting system R and consider GWP in <italic>G</italic> which is defined by GWP(w,U) iff w∈ <U> for w∈ <italic>G</italic>, finite U⊆<italic>G</italic>, where <U> is the subgroup of <italic>G</italic> generated by U. We describe <U> also by a rewrite system S and define a rewrite relation @@@@<subscrpt>S,R</subscrpt> in such a way that w @@@@<subscrpt>S,R</subscrpt> λ iff w∈ <U> (λ the empty word). For this rewrite relation we develop different critical pair criteria for @@@@<subscrpt>S,R</subscrpt> to be λ-confluent, i.e. confluent on the left-congruence class [λ] of @@@@<subscrpt>S,R</subscrpt>. Using any of these λ-confluence criteria we construct a completion procedure which stops for every input S and computes a λ-confluent rewrite system equivalent to S. This leads to a decision procedure for GWP in <italic>G</italic>. Thus we give an explicit uniform algorithm for deciding GWP in polycyclic groups and a new proof based almost only on rewriting techniques for the decidability of this problem. Further, we define a rewrite relation @@@@<subscrpt>LM,U</subscrpt> which is stronger than @@@@<subscrpt>S,R</subscrpt>. We show that if <italic>G</italic> is given by a nilpotent string-rewriting system, then by a completion procedure the input U can be transformed into V such that @@@@<subscrpt>LM,V</subscrpt> is even confluent, not just λ-confluent.