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
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.
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