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
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