This phenomenon can be understood as follows:    

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive **[hyperbolic group][1]** (and non virtually cyclic).  

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ (and  orbit of (0,0)), the  **[Cayley graph][2]** of $\Gamma$ (Gromov hyperbolic geometry)  is projected (*in a manner not too degenerate*) on $\mathbb{Z}^2$ (euclidean geometry).      
Then the emergence of explosive and **[fractal][3]**-like structures on the spheres is not surprising.    
  
So any (non virtually cyclic) hyperbolic group would generate beautiful fractal-like pictures.   



**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 $13$:  

![enter image description here][4]   

  
**Remark**:  I don't know if your group $G$ is hyperbolic, but we could have the same understanding for all the finitely generated groups of exponential growth.  


  [1]: http://en.wikipedia.org/wiki/Hyperbolic_group
  [2]: http://en.wikipedia.org/wiki/Cayley_graph
  [3]: http://en.wikipedia.org/wiki/Fractal
  [4]: https://i.sstatic.net/S8rVw.png