Analysing Finitely Presented Groups by Constructing Representations

One idea how to prove that a finitely presented group G is infinite is to construct suitable homomorphisms into infinite matrix groups. In [HoP 92] this is done by starting with a finite image H of G and solving linear equations to check whether the epimorphism onto H can be lifted to a representation whose image is an extension of a ZZ-lattice by H, thus exhibiting an infinite abelian section of G in case it exists near the top of the group. The variation of the idea presented here seems to be suitable for instance for some groups where the kernel of the epimorphism onto H has a pro-p-completion of bounded width. The price to be paid is that algebraic rather than linear equations have to be solved. These are obtained from evaluating the relations on matrices with indeterminates as entries. This system of equations can be simplified by applying some representation theory, but it often remains too complicated for direct methods e.g. Gröbner base calculations or elimination of variables by taking resultants. An alternative to the direct methods is to find a modular solution and to lift it to a solution over a p-adic field. Knowing a finite epimorphic image H of G, H and its modular representation theory might give us a hint which modular representation of H one should try to lift to a p-adic representation of G. A multi-dimensional version of Hensel’s lemma often assures that it is sufficient to lift a solution only a few steps to conclude that it can be lifted to a solution over a p-adic field. Note, although the initial equations are algebraic, each step of the lifting process is just a problem of solving linear equations over a finite field. In general it is not clear that the lifting process leads to entries which are algebraic over I Q even in the situation where each lift is equivalent to a representation whose entries are algebraic over I Q. To get a representation defined over a number field K (with degree [K : I Q] as small as possible), therefore involves two steps: To pass from the lifted representation to one with p-adic entries which are algebraic over I Q and to identify the new entries as algebraic numbers. In the examples of this paper the first step turned out to be superfluous, since the lifted representation had already entries with rational minimal polynomials of small degree, see however [HPS 94] for a more ill-behaved example. The second step is discussed in Section 5. Sometimes the lift does not work in characteristic 0, however is possible in the ring IFp[[x]] of Laurent series over IFp. An even simpler method than the one for number fields yields a representation over the field of rational functions over IFp. In both