In this thesis we use non-crossing partitions(NCP) to examine Artin groups of finite type and their subgroups. We follow the work of Brady-Watt and Bessis to construct a con- tractible universal cover for the Artin group B(W) using this NCP structure. This space X can be factored out by normal subgroups H of B(W) and the resulting quotient space is a K(H,1). We examine this quotient space to see what information it gives us about the subgroup H.
The main result of this thesis is that B(F4)′, the commutator subgroup of the Artin group B(F4), is finitely presented. It is already known whether the commutator subgroups of the other irreducible Artin groups are finitely presented. We retract the space X in the F4 case and filter it appropriately to apply a theorem of Brown. If the filtration is finite mod B(F4)′ and successive stages of the filtration are obtained from the previous stage by the adjunction of 3-cells then B(F4)′ has finiteness type F2 but not finiteness type F3.
We also recover the fact that B(C3)′ is finitely generated but not finitely presented. This is done by examining the fundamental group and second homology group of our K(B(C3)′, 1).
The other subgroup we are interested in is the kernel of the map which sends the NCP generators of an Artin group to the lengths of the corresponding non-crossing partitions. We define a Morse function on the quotient space in this case to calculate the homology. The Morse function on the quotient space also defines one on truncations of the NCP lattice. The simplification resulting from this Morse function recovers the fact that the homology of these truncations is entirely in the top dimension.