Large mapping tori of free group endomorphisms

We present an algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We use this to prove that a mapping torus of a finitely generated free group automorphism is large if it contains the integers times the integers as a subgroup of infinite index. We then extend this result to mapping tori of finitely generated free group endomorphisms, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We also show that if a group possesses a deficiency 1 presentation where one of the relators is a commutator then it is the integers times the integers, or it is large, or it is as far as possible from being residually finite.