Tile Homotopy Groups

Tile Homotopy Groups, by Michael Reid
L'Enseignement Mathématique 49 (2003), no. 1-2, pp. 123-155.
[DOI] [Math Reviews] [Zentralblatt]
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The technique of using checkerboard colorings to show impossibility of some tiling problems is well-known. Conway and Lagarias [1] have introduced a new technique using boundary words. They show that their method is always at least as strong as any generalized coloring argument, and in some cases, is strictly stronger. They successfully apply their technique, which involves some understanding of specific finitely presented groups, to two tiling problems. Partly because of the difficulty in working with finitely presented groups, their technique has only been applied in a handful of cases.
We present a slightly different approach to the Conway-Lagarias technique, which we hope provides further insight. We also give a strategy for working with the finitely presented groups that arise. Many new examples are given where we can apply this strategy successfully.
[1] J.H. Conway and J.C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, Journal of Combinatorial Theory, Series A 53 (1990), no. 2, pp. 183-208. [Math Reviews] [Zentralblatt]
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Updated February 5, 2010.