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It is known that word-hyperbolic groups admit a finite K(\pi,1) and have no `Baumslag-Solitar' subgroups, but it is not known whether the converse is true. We will discuss this question for mapping tori of injective free group endomorphisms, and prove that the absence of Baumslag-Solitar subgroups is equivalent to being word-hyperbolic for a large class of endomorphisms. We will also discuss when the non-hyperbolic mapping tori admit a quadratic isoperimetric inequality.