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

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http://dx.doi.org/10.1090/cbms/117/01