Mark Hagen

Cutting up free-by-cyclic groups

Fix a finite-rank free group F and an automorphism f of F, and let G be the mapping torus of f. I will start by recalling how G is classified according to the growth rate of f, and explain the three structural results that underpin much of our understanding of the geometry of G: the relative hyperbolicity result by Dahmani-Li and Ghosh, the cyclic hierarchy originating in work of Macura, and the "JSJ-type'' decomposition due to Andrew-Martino and, independently, Dahmani-Touikan. I will then attempt to motivate the question of when G acts freely on a CAT(0) cube complex and mention some partial results (very old, and also very recent, joint work with Wise). Free cocompact actions on cube complexes are impossible for many G, by recent work of Munro-Petyt. I will conclude with a result that uses the work of Munro-Petyt to produce an algebraic sufficient condition on G ruling out a quasi-isometry to a CAT(0) cube complex.