Finding indivisible Nielsen paths for a train track map

Summary

Abstract

An algorithm is described for determining the indivisible Nielsen paths for a train track map and therefore the subgroup of elements of the fundamental group fixed by the induced automorphism.

In my talk at the Edinburgh Conference I described the “procedure” for finding fixed points of an automorphism of a free group that is implicit in [6] and explicit in [2] and [7]. I made the point that this is a procedure and not an algorithm since there is no way in general of knowing how long to persist before being sure that all fixed elements have been found—although it has been shown to be effective for positive automorphisms [2]. The example of Stallings [8], p99, figure 3 (Example 1 below) was presented to show that one may need more persistence than expected. I also discussed the notion of an indivisible Nielsen path (INP) which was introduced in [1] and used as a fundamental tool in [4]. After the talk, Bestvina asked me whether the procedure could be adapted to determine the INPs for a train track map, the determination of which is an essential part of the algorithm introduced in [1] for finding the fixed words of the induced automorphism. This paper shows how to do this. All irreducible automorphisms have train track representatives so this provides a straightforward means (given the train track map!) of computing the generator of the fixed subgroup.