Non Hyperbolic Free-By-Cyclic and One-Relator Groups

We show that the free-by-cyclic groups of the form F(2)-by-Z act properly cocompactly on CAT(0) square complexes. We also show using generalised Baumslag-Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to BS(1,j). Finally we consider free-by-cyclic groups which are not relatively hyperbolic with respect to any collection of subgroups.

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