CHAPTER 1
Overview
Our goal is to describe a stream of geometric group theory connecting
many of the classically considered groups arising in combinatorial group
theory with right-angled Artin groups. The nexus here are the “special
cube complexes” whose fundamental groups embed naturally in right-angled
Artin groups.
Nonpositively curved cube complexes, which Gromov introduced merely
as a convenient source of examples [28], have come to take an increasingly
central status within parts of geometric group theory especially among
groups with a comparatively small number of relations. Their ubiquity is
explained by Sageev’s construction [65] which associates a dual cube com-
plex to a group that has splittings or even ‘semi-splittings’ i.e. codimension-1
subgroups.
Right-angled Artin groups, which at first appear to be a synthetic class
of particularly simple groups, have taken their place as a natural target
possibly even a “universal receiver” for groups that are well-behaved and
that have good residual properties and many splittings or at least “semi-
splittings”.
We begin by reviewing nonpositively curved cube complexes and a disk
diagram approach to them first entertained by Casson (above dimension 2).
These disk diagrams are used to understand their hyperplanes and convex
subcomplexes. While many of the essential properties of CAT(0) cube com-
plexes can be explained using the CAT(0) triangle comparison metric, we
have not adopted this viewpoint. It seems that the most important charac-
teristic properties of CAT(0) cube complexes arise from their hyperplanes,
and these are exposed very well through disk diagrams and the view will
serve us further when we take up small-cancellation theory.
Special cube complexes are introduced as cube complexes whose im-
mersed hyperplanes behave in an organized way and avoid various forms
of self-intersections. CAT(0) cube complexes are high-dimensional general-
izations of trees, and likewise, from a certain viewpoint, special cube com-
plexes play a role as high-dimensional “generalized graphs”. In particular
they allow us to build (finite) covering spaces quite freely, and admit natural
virtual retractions onto appropriate “generalized immersed subgraphs” just
like ordinary 1-dimensional graphs. The fundamental groups of special cube
complexes embed in right-angled Artin groups because of a local isometry
to the cube complex of a naturally associated raag. Since right-angled Artin
1
http://dx.doi.org/10.1090/cbms/117/01
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