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Title: A Coarse Geometric Obstruction to Cubulation

Abstract: A major theme of geometric group theory over the past few decades has been to cubulate groups, i.e., construct proper cocompact actions on CAT(0) cube complexes. Such actions often yield algebraic consequences (e.g. subgroup separability, biautomaticity, aTmenability), and a group is often obstructed from cubulation by not having one of these algebraic properties. Taking a different, geometric point of view, we define a class of 'poisonous' spaces -- richly branching flats (RBFs) -- and we show that groups containing RBFs are not quasi-isometric to CAT(0) cube complexes and thus not cubulated. We present some applications to free-by-cyclic groups and tubular groups. This work is joint with Harry Petyt.