We introduce and examine a special class of cube complexes. We show that special cube-complexes virtually admit local isometries to the standard 2-complexes of naturally associated right-angled Artin groups. Consequently, special cube-complexes have linear fundamental groups. In the word-hyperbolic case, we prove the separability of quasiconvex subgroups of fundamental groups of special cube-complexes. Finally, we give a linear variant of Rips's short exact sequence.
We study the B(6) and B(4)-T (4) small cancellation groups. These classes include the usual C (1/6) and C (1/4)-T (4) metric small cancellation groups. We show that every finitely presented B(4)-T (4) or word-hyperbolic B(6) group acts properly discontinuously and cocompactly on a CAT(0) cube complex. We show that finitely generated infinite B(6) and B(4)-T (4) groups have codimension 1 subgroups and thus do not have property (T). We show that a finitely presented B(6) group is wordhyperbolic if and only if it contains no Z × Z subgroup.
We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol's criterion, we find that every subgroup separable closed hyperbolic 3-manifold is virtually fibered.
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