Mapping tori of small dilatation expanding train-track maps

An expanding train-track map on a graph of rank n is P-small if its dilatation is bounded above by Pn. We prove that for every P there is a finite list of mapping tori X1,…,XA, with A depending only on P and not n, so that the mapping torus associated with every P-small expanding train-track map can be obtained by surgery on some Xi. We also show that, given an integer P>0, there is a bound M depending only on P and not n, so that the fundamental group of the mapping torus of any P-small expanding train-track map has a presentation with less than M generators and M relations. We also provide some bounds for the smallest possible dilatation.