This phenomenon can be understood as follows:
Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).
Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ and the orbit of $(0,0)$, the Cayley graph (Gromov hyperbolic geometry) of $\Gamma$ is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
So the emergence of (explosive and) fractal-like structures on the spheres is not surprising. The same computation with several such hyperbolic groups would generate many beautiful fractal-like pictures.
I don't know if your group $G$ is hyperbolic, but we can have the same understanding.
Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that
- $a: (m,n) \to (m+1,m+n)$
- $b: (m,n) \to (m+n,n+1)$
To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$, and is isomorphic to the free group $\mathbb{F}_2$ (so $\Gamma$ is hyperbolic).
The cardinalities of the spheres of radii $r=0,\dots,13$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720$
The entire sphere of radius $13$:
