This is an idea I'm sure exists already, but is quite complicated, so it's hard to find without appropriate terminology. We consider a tiling of the plane by regular $n$-gons, containing an edge $e_0$ from $(0,0)$ to $(1,0)$. We are interested in the set of "oriented edges" of this tiling. These are ordered pairs $(e,P)$ where $e$ is an edge of the tiling, and $P$ is a polygon in the tiling incident to $e$. The reason we're interested in these objects, more complicated that mere edges, is that they have nicer properties for our purposes, as will become apparent.
We will call the polygon incident to $e_0$ contained in the upper half-plane $\{(x,y)\ |\ y\geq0\}$ by the name $P_0$. We begin with the oriented edge $(e_0,P_0)$. Each edge $e$ of $P_0$ is incident to another polygon, $P$. There is a unique oriented isometry of $\mathbb{R}^2$ mapping $e$ onto $e_0$ and $P$ onto $P_0$. This lets us think of each oriented edge of $P$ as an edge of $P_0$, if we indicate this isometry by $e$. That is, each oriented edge $(x,P)$ of $P$ can be thought of as finite sequence $ef$ where $f$ is another edge of $P_0$ corresponding to $x$ under the isometry. But $x$ is incident to another polygon $Q$, and there is a unique oriented isometry taking $x$ to $e_0$ and $Q$ to $P_0$. So we may think of an oriented edge $(y,Q)$ of $Q$ as a finite sequence $efg$ of edges of $P_0$. With this method, every oriented age is associated to a finite sequence of edges, although not uniquely. $(e_0,P_0)$ is associated to the empty sequence, among others.
This operation is reversible. If we have $efg...kl$ giving $(x,P)$, obtaining $(y,Q)$ as $efg...k$ is as simple as finding the unique edge $y$ of $P$, such that $y$ is incident to $P$ and $Q$, and the oriented isometry sending $y$ to $e_0$ and $Q$ to $P_0$ maps $x$ onto $l$. This operation can be applied to any oriented edge, so we'll call it $l^{-1}$. In this way, we have associated every member of the free group $F_n$ generated by the edges of $P_0$ to an edge of the tiling surjectively, although not injectively.
Now the members of $F_n$ corresponding to $(e_0,P_0)$ form a normal subgroup $N$ of $F_n$. This is why we used oriented edges. Is there a nice description of the members of $N$, an isomorphism between $N$ and a nice group or an isomorphism between $\frac{F_n}{N}$ and a nice group.