A Group of Paths in R

We define a group structure on the set of compact “minimal” paths in R2. We classify all finitely generated subgroups of this group G: they are free products of free abelian groups and surface groups. Moreover, each such group occurs in G. The subgroups of G isomorphic to surface groups arise from certain topological 1-forms on the corresponding surfaces. We construct examples of such 1-forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using G we construct a non-polygonal tiling problem in R2, that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group G has applications to combinatorial tiling problems of the type: given a set of tiles P and a region R, can R be tiled by translated copies of tiles in P ?