Representations, commutative algebra, and Hurwitz groups Dedicated to Charles Leedham-Green on the occasion of his 65th birthday

The developed methods are much more general and can in principle be used to construct representations of any finitely presented group. One assigns matrices with indeterminate entries to the generators of the group so that the group relations become relations between commuting variables, as was already suggested in [PlS 97] and applied in [HPS 97]. Meanwhile rather effective methods for treating algebraic equations are developed so that it is worthwhile to come back to the old topic. In particular, we use an effective implementation of Janet’s algorithm, an earlier and rather successful version of a Gröbner basis type algorithm, cf. [BCG 03], [BGY 01], [PlR 05], to deal with the equations. For the relevance of Hurwitz groups in the context of mathematics in general the reader is referred to [Mac 99], for its group theoretical relevance and history to [TaV 06].