In this paper and its sequel we present a method that, under loose restrictions, is algorithmic for calculating the Nielsen type numbers NΦₙ(f) and NPₙ(f) of self maps f of hyperbolic surfaces with boundary and also of bouquets of circles. Because self maps of these surfaces have the same homotopy type as maps on wedges of circles, and the Nielsen periodic numbers are homotopy type invariant, we need concentrate only on the latter spaces. Of course the results will then automatically apply to the former spaces as well. The algorithm requires only that f has minimal remnant, by which we mean that there is limited cancellation between the f⁎ images of generators of the fundamental group. These methods often work even when the minimal remnant condition is not satisfied.
Our methodology involves three separate techniques. Firstly, beginning with an endomorphism h on the fundamental group, we adapt an algorithm of Wagner to our setting, allowing us to distinguish non-empty Reidemeister classes for iterates of a special representative map for h, which we introduce. Secondly, using techniques reminiscent of symbolic dynamics, we assign key algebraic information to the actual periodic points of this special representative. Finally, we use word length arguments to prove that the remaining information required for the calculation of NΦₙ(f) and NPₙ(f) can be found with a finite computer search. We include many illustrative examples.
In this first paper we give the tools we need in order to present and give the algorithm for NPₙ(f). All the tools introduced here will be needed in the sequel where we develop the extra tools needed in order to compute NΦₙ(f).