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Title: Train tracks, orbigraphs, and CAT(0) free-by-cyclic groups.

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Abstract:听Given听蠁:Fn鈫扚n蠁:Fn鈫扚n听an automorphism of a free group of rank听nn, there is an associated free-by-cyclic group听Fn鈰娤哯Fn鈰娤哯, which may be thought of as the mapping torus of the automorphism. Properties of the automorphism determine properties of the mapping torus and vice-versa. Gersten gave a simple example听蠄:F3鈫扚3蠄:F3鈫扚3听of an automorphism whose mapping torus is a "poison subgroup" for nonpositive curvature, in the sense that any group containing听F3鈰娤圸F3鈰娤圸听is not a CAT(0) group. In the opposite direction, Hagen-Wise and Button-Kropholler proved certain families of automorphisms have mapping tori that are cocompactly cubulated. We prove that a large class of polynomially-growing free group automorphisms admitting an additional symmetry have CAT(0) mapping tori. The key tool is a representation of these automorphisms as relative train track maps on听orbigraphs, certain graphs of groups thought of as orbi-spaces. This gives a听hierarchy听for the mapping torus. It is an interesting question whether or not our mapping tori are cocompactly cubulated.