Cayley Graphs on Surfaces
Some aspects may be related:
curvature of cayley graph
curavture of surface (The convergence result that for n going to infinity, the Ollivier-Ricci curvature of the random geometric graph sampled from manifold converges to the Ricci curvature of the manifold) (and for more general theory, see Graphs of Groups on Surface: Interactions and Models by Arthur T. White).
embedding of cayley graph on surface (see “Cayley maps” by R. Bruce Richter et al.)
We can do an experiment, followed by a statistical analysis: do Poisson process to verify the convergence result experimentally and apply to real data cloud on a manifold to observe clustering/communities and do learning etc.