Analysing Finitely Presented Groups by Constructing Representations

We present methods to construct representations of finitely presented groups. In well-conditioned examples it is possible to use Gröbner base and resultant methods to solve the system of algebraic equations obtained by evaluating the relations on matrices with indeterminates as entries. For more complicated cases we show how finite epimorphic images and their modular representation theory can be used to find good candidates for representations over a finite field that can be lifted to a local field. Finally, we apply lattice reduction methods to representations over local fields and obtain representations over algebraic number fields resp. rational function fields. Reducing modulo prime ideals thus gives infinitely many images of the finitely presented group.