Event

Rylee Lyman (Tufts University)

Wednesday, January 29, 2020 15:00to16:00
Burnside Hall BURN 1104, 805 rue Sherbrooke Ouest, Montreal, QC, H3A 0B9, CA

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

 

Abstract: Given φ:Fn→Fnφ:Fn→Fn an automorphism of a free group of rank nn, there is an associated free-by-cyclic group Fn⋊φZFn⋊φZ, 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 ψ:F3→F3ψ:F3→F3 of an automorphism whose mapping torus is a "poison subgroup" for nonpositive curvature, in the sense that any group containing F3⋊ψZF3⋊ψZ 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 orbigraphs, certain graphs of groups thought of as orbi-spaces. This gives a hierarchy for the mapping torus. It is an interesting question whether or not our mapping tori are cocompactly cubulated.

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