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A group of paths in $$\mathbb{R}^2$$. (English) Zbl 0896.52024
Summary: 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. Morover, 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 $$P$$?

##### MSC:
 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 20E08 Groups acting on trees 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 05B45 Combinatorial aspects of tessellation and tiling problems
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