We show that certain right-angled Coxeter groups have finite index subgroups that quotient to Z with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in H 4 with fundamental domain the 120-cell or the 24-cell.

For each n we construct examples of finitely presented C ′ (1/6) small cancellation groups that do not act properly on any n-dimensional CAT(0) cube complex.

We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group A to be (virtually) cocompactly cubulated, in terms of the defining graph of A.

We show that many 2-dimensional Artin groups are residually finite. This includes 3-generator Artin groups with labels ≥ 3 where either at least one label is even, or at most one label is equal 3. As a first step towards residual finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed.

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