COMPUTING PRESENTATIONS FOR SUBGROUPS OF POLYCYCLIC GROUPS AND OF CONTEXT-FREE GROUPS

Algorithms for solving uniform decision problems for algebraic structures crucially depend on the chosen finite presentations for the structures under consideration. Rewriting techniques have been used very successfully to solve uniform decision problems, when the presentations considered involve finite, noetherian, and (lambda)-confluent rewriting systems. Whenever the class of algebraic structures considered is closed under the operation of taking finitely generated substructures, then the algorithms for solving the uniform decision problems can be applied to the sub-structures as well. However, since these algorithms depend on the form of the presentations, this involves the task of constructing a presentation of a certain form for a substructure given a presentation of this form for the structure itself and a finite set of generating elements for the substructure. This problem, which has received a lot of attention in algebra, is here investigated from an algorithmic point of view. The structures considered are the following two classes of groups, which have been studied extensively before: the polycyclic groups and the context-free groups. Finitely generated context-free groups can be presented by finite, monadic, and lambda-confluent string-rewriting systems. Due to their nice algorithmic properties these systems provide a way to effectively solve many decision problems for context-free groups. Since finitely generated subgroups of context-free groups are again context-free, they can be presented in the same way. Here we describe a process that, from a finite, monadic, and lambda-confluent string-rewriting system presenting a context-free group G and a finite subset U of G, determines a presentation of this form for the subgroup [U] of G that is generated by U. For finitely presented polycyclic groups we obtain an analogous result, when we use finite confluent PCP2-presentations to describe these groups.