| Let {dollar}R{dollar} be a number field. We present several algorithms for working with polycyclic-by-finite subgroups of GL({dollar}n, R{dollar}). Let {dollar}G{dollar} be a subgroup of GL({dollar}n, R{dollar}) given by a finite generating set of matrices. We describe an algorithm for deciding whether or not {dollar}G{dollar} is polycyclic-by-finite. For polycyclic-by-finite {dollar}G{dollar}, we describe an algorithm for deciding whether or not a given matrix is an element of {dollar}G{dollar}.; We prove that an abstract group {dollar}G{dollar} has a faithful representation as a triangularizable subgroup of GL({dollar}n{dollar}, Z) for some {dollar}n{dollar} if and only if {dollar}G{dollar} is polycyclic and the commutator subgroup of {dollar}G{dollar} is torsion-free nilpotent. Suppose {dollar}G{dollar} is a polycyclic group given by a consistent polycyclic presentation. We describe an algorithm for deciding whether or not {dollar}G{dollar} has a faithful representation as a triangularizable subgroup of GL({dollar}n{dollar}, Z), as well as an algorithm for constructing such a representation if it exists.; Preliminary experiments indicate that the algorithms described in this thesis are suitable for computer implementation. Further experimentation is needed to determine the range of input for which the algorithms are practical with current technology. |
