Deciding finiteness for matrix groups over function fields

Let F be a field and t an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n, F(t)) is finite. When F is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n, F), for which the results of [1] can be applied. When F is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Well and upper bounds on the size of a finite subgroup generated by a fi xed number of generators with examples of constructions almost achieving th e bounds. We use these results to then give exponential deterministic algor ithms for deciding finiteness as well as some preliminary results towards m ore efficient randomized algorithms.