A group of paths in $\mathbb {R}^2$

Abstract:

We define a group structure on the set of compact “minimal” paths in $\mathbb {R}^2$. 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 $\mathbb {R}^2$, 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 $P$?