Picture groups of finite type and cohomology in type $A_n$

For every quiver (valued) of finite representation type we define a finitely presented group called a picture group. This group is very closely related to the cluster theory of the quiver. For example, positive expressions for the Coxeter element in the group are in bijection with maximal green sequences [IT17]. The picture group is derived from the semi-invariant picture for the quiver. We use this picture to construct a finite CW complex which (by [IT16]) is a $K(\pi,1)$ for this group. The cells are in bijection with cluster tilting objects. For example, in type $A_n$ there are a Catalan number of cells. The main result of this paper is the computation of the cohomology ring of all picture groups of type $A_n$ with any orientation and any coefficient ring.