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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