I am studying (on my own) some random group theory, and using this primer. The book focuses on finitely presented groups, and the main definition of a hyperbolic group there is "word-hyperbolic", meaning: given a word $w$ representing the trivial element in the group, the minimal number of relators needed to write $w$ is bounded by some constant times its length $|w|$ (see page 18).
In page 20 the author explains the connection between tilings and groups, but he does so very informally. Specifically, he says that the hyperbolic group $\langle a,b,c,d \mid aba^{-1}b^{-1} cdc^{-1}d^{-1} \rangle$ corresponds to octagon tiling of the hyperbolic plane (tiles correspond to the relation), and that because there's a tiling of the Euclidean plane with hexagons, there's a group which satisfies the $C'(1/6+\varepsilon)$ condition without being hyperbolic.
I understand the intuition - the area of hexagon tiling grows linearly while the perimeter quadratically, and in hyperbolic octagon they both grow at the same rate. Still, the correspondence lacks "explicitation" for me.
My questions are:
- Can we generally construct hyperbolic (resp. non-hyperbolic) groups from tiling of hyperbolic (resp. non-hyperbolic) space? My first guess would be to take the symmetry group of a tiling, but will it Cayley graph be really similar to the tiling? Will it be hyperbolic if we started with hyperbolic space?
- Suppose I take the group $\langle S \mid R \rangle$ where $R$ is a set of one relation of length 6. When will it necessarily be non-hyperbolic (in the sense of word-hyperbolic)? For example consider $S=\{ abcabc \}$, $S= \{ aba^{-1} cdc^{-1}\}$ or $S= \{abc a^{-1} b^{-1} c^{-1} \}$. (EDIT: I'm pretty confident that the last example is indeed non-hyperbolic, since I chose the generator to give a valid tiling)
Hyperbolicity, as I understand it, is a property of a finitely presented group that only depends on its Cayley graph as a metric space up to quasi-isometry. Consequently, you can study it by studying any metric space quasi-isometric to the Cayley graph. An important theorem is that if $M$ is a closed Riemannian manifold, then $\pi_1(M)$ is quasi-isometric to the universal cover $\widetilde{M}$. In particular, if $M$ is a closed hyperbolic surface, then $\pi_1(M)$ is quasi-isometric to the hyperbolic plane. $\widetilde{M}$ is tiled by copies of $M$ so that's where the tiling comes into play.