L'Enseignement Mathématique
(2003), no. 1-2, pp. 123-155.

Tile Homotopy Groups,
by Michael Reid

L'Enseignement Mathématique
(2003), no. 1-2, pp. 123-155.

Abstract

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.

Reference

[1]
J.H. Conway and J.C. Lagarias,
Tiling with Polyominoes and Combinatorial Group Theory,
Journal of Combinatorial Theory, Series A
(1990), no. 2, pp. 183-208.
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Updated February 5, 2010.