  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.
