Acknowledgement I am enormously grateful to Jason Behrstock for organizing the NSF- CBMS conference together with Abhijit Champanerkar. I am also grateful to the conference participants for their feedback, to the CUNY Graduate Center for hosting the conference, the CBMS for choosing this topic, and the NSF for funding it. My research has been supported by NSERC and undertaken at McGill University. During my 2008-2009 sabbatical at the Hebrew University I benefited tremendously from the feedback and encouragement of Zlil Sela, with whom I went through the project on The Structure of Groups with a Quasiconvex Hierarchy that led to this conference. Valuable criticism and helpful cor- rections were subsequently provided by many other friends and colleagues, among them Ian Agol, Nicolas Bergeron, Hadi Bigdely, Collin Bleak, David Gabai, Mark Hagen, Jason Manning, Eduardo Martinez-Pedroza, Daniel Moskovitch, Piotr Przytycki, and Eric Swenson. I am grateful to Mar- tin Bridson for initiating my interest in nonpositively curved square com- plexes. My daughter Talia helped with the figures and my parents Batya and Michael helped me with crayons and fractions respectively, and have been a source of continuing and enthusiastic support. It is a great fortune to have wonderful collaborators who spent much time and energy working with me on the ideas presented here: Tim Hsu on linearity, separability in right-angled Artin groups, and cubulating mal- normal amalgams Chris Hruska on the finiteness properties of the dual cube complex, the bounded packing property, and the tower approach to the spelling theorem for one-relator groups with torsion. I’ve engaged with Michah Sageev’s seminal work on CAT(0) cube complexes for the last 10 years and have learned many things working with him. My long-term col- laboration with Fr´ ed´ eric Haglund on special cube complexes is at the heart of the matter here, and I am additionally building on his influential earlier ideas. Finally, I am grateful to my adorable and loving wife Yael for giving me the time to write these notes, and for serving as a partner, a cheerleader, and a role model in all parts of my life. xi
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