A group of paths in

We define a group structure on the set of compact ``minimal' paths in . We classify all finitely generated subgroups of this group : they are free products of free abelian groups and surface groups. Moreover, each such group occurs in . The subgroups of isomorphic to surface groups arise from certain topological -forms on the corresponding surfaces. We construct examples of such -forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using we construct a non-polygonal tiling problem in , that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group has applications to combinatorial tiling problems of the type: given a set of tiles and a region , can be tiled by translated copies of tiles in ?