Algorithms for Polycyclic-by-finite Matrix Groups

Abstract

Let R be the ring of integers or a number field. We present several algorithms for working with polycyclic-by-finite subgroups of GL(n; R). Let G be a subgroup of GL(n; R) given by a finite generating set of matrices. We describe an algorithm for deciding whether or not G is polycyclic-by-finite. For polycyclic-by-finite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We also describe an algorithm for deciding whether or not G is solvable-by-finite, providing an alternative to the algorithm proposed by Beals ([Be1]) for this problem. Baumslag, Cannonito, Robinson and Segal prove that the problem of determining whether or not a finitely generated subgroup of GL(n; Z) is polycyclic-by-finite is decidable and that the problem of testing membership in a polycyclic-by-finite subgroup of GL(n; Z) is also decidable ([BCRS]). In this report we extend these results by describing algorithms which appear to be suitable for computer implementation. Exper..