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Cubical small-cancellation theory and large-dimensional hyperbolic groups https://doi.org/10.17863/CAM.101703 Arenas, Macarena 0000000349658121 Wilton, Henry Given a finitely presented group Q and a compact special cube complex X with nonelementary hyperbolic fundamental group, we produce a non-elementary, torsion-free, cocompactly cubulated hyperbolic group Γ that surjects onto Q, with kernel isomorphic to a quotient of G = π_1X and such that max{cd(G),2} ≥ cd(Γ) ≥ cd(G)−1. Separately, we show that under suitable hypotheses, the second homotopy group of the coned-off space associated to a C(9) cubical presentation is trivial, and use this to provide classifying spaces for proper actions for the fundamental groups of many quotients of square complexes admitting such cubical presentations. When the cubical presentations satisfy a condition analogous to requiring that the relators in a group presentation are not proper powers, we conclude that the corresponding coned-off space is aspherical. University of Cambridge 2023-05-15 Thesis Doctoral Doctor of Philosophy (PhD) eng Cambridge Trust & Newnham College scholarship https://www.repository.cam.ac.uk/handle/1810/357537 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f6a15b5c-4dcb-4644-8070-3e579384a658/download 0705a1a26770f70ca7b60e48c53c4764 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/1915c927-9913-48ee-a6af-b0b15e6f08b3/download 87eda9de84448d1f82354d60eee3eb5f https://creativecommons.org/licenses/by/4.0/ hyperbolic groups cube complexes small-cancellation theory