This research is in a rapidly developing area that blends algebra with computer science. It supports widely varied applications in mathematics and further afield. This area, still very new, concerns symbolic computation with finitely generated linear (matrix) groups over infinite rings. Our main objective is to continue the development of techniques that exploit Zariski density and residual finiteness of input groups. The main focus is new techniques for isometry groups over S-integral domains contained in number fields. There is demand for practical methods to compute with such groups, notably from researchers outside group theory (algebraic geometers, topologists, and number theorists). All planned algorithms will be implemented. They will comprise the first software of this kind to become available for general use. Through experimentation with our software, we expect significant progress towards solution of long-standing open problems; for example, (non)-freeness testing of parabolic Möbius groups.