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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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