Consider V(χ,r) at base point χ.... Consider shells formed at particular radial distances from a given base point....
Generalize spherical harmonics:
FunctionExpand[SphericalHarmonicY[3/2,1/2,θ,ϕ]]
In[]:=
ϕ
2

1/4
(1+Cos[θ])
(-1+2
2
Cos[θ]
)
Sin[θ]

2
π
1+
1
2
(-1+Cos[θ])
1/4
(1-Cos[θ])
1/4
(1-
2
Cos[θ]
)
Out[]=
FunctionExpand[SphericalHarmonicY[5/3,1/2,θ,ϕ]]
In[]:=
ϕ
2

1/4
(1+Cos[θ])
Cos
13
3
ArcSin
1-Cos[θ]
2

Sin[θ]

2
π
1+
1
2
(-1+Cos[θ])
1/4
(1-Cos[θ])
1/4
(1-
2
Cos[θ]
)
Out[]=
FunctionExpand[SphericalHarmonicY[l,m,θ,ϕ]]
In[]:=
mϕ

1+2l
-m/2
(1-Cos[θ])
m/2
(1+Cos[θ])
-m/2
(1-
2
Cos[θ]
)
Gamma[1+l-m]
Hypergeometric2F1-l,1+l,1-m,
2
Sin
θ
2


m
Sin[θ]
2
π
Gamma[1-m]
Gamma[1+l+m]

Out[]=
“Growth function”
https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf

Model spaces

Start from an ID for every point; ID is a sequence of symbols (e.g. the sequence of rule IDs that produced that node)
Now need to identify lots of ID sequences in a “uniform” way.
KaryTree[300]
In[]:=
Out[]=
Every node could be the root.
Given the ID, which is an infinite sequence of 0s and 1s, split it anywhere and consider it representing up or down on the tree.
On a tree (i.e. free group), you can get to a new element by appending any sequence of generators.
[ Is model space the universal covering space ]
Is the Sierpinski the Cayley graph of some infinite group?