Special project: Tiling with groups

A good reference for this section is Thurston's article [Th].

Let $P$ be any oriented path on a rectangular lattice. Let $a,b$ be any elements of some group $G$ . We associate to $P$ a ``word'' in $a,b$ as follows. For each move $\rightarrow$ by one unit in $P$ , write $a$ . For each move $\leftarrow$ by one unit in $P$ , write $a^{-1}$ . For each move $\uparrow$ by one unit in $P$ , write $b$ . For each move $\downarrow$ by one unit in $P$ , write $b^{-1}$ .

Example 4.9.1   Let $P$ be the closed path pictured below, where $\bullet$ denotes the initial (and terminal) point.

The word associated to $P$ is

$$w(P)=a^3ba^{-1}ba^{-1}b^{-1}a^{-1}b^{-1}.$$

Exercise 4.9.2   Let $w_P(a,b)$ be the word associated to a path $P$ . Let $-P$ denote the path associated to the path $P$ with every edge having the opposite orientation. Show $w_{-P}(a,b)=w_P(a,b)^{-1}$ .

Exercise 4.9.3   Conside the tile in Example 4.9.1. Find the words associated to each of its rotations and reversals.

Exercise 4.9.4   Show that the square $4\times 4$ grid can be tiled using the tile in Example 4.9.1 along with its rotations and reversals.

Exercise 4.9.5   Using group theory, show that the square $10\times 10$ grid cannot be tiled using the tile in Example 4.9.1 along with its rotations and reversals.

Exercise 4.9.6   Show that the $4\times 4$ square with missing upper left-hand and lower right-hand corners

cannot be tiled by the $1\times 2$ tile

along with rotations by multiples of $90^o$ and reversals.