  Guided by physical needs, we deal with the rotationally isotropic Poincar\'e
ball, when considering the complement of Borromean rings embedded in it. We
consistently describe the geometry of the complement and realize the
fundamental group as isometry subgroup in three dimensions. Applying this
realization, we reveal normal stochastization and multifractal behavior within
the examined model of directed random walks on the rooted Cayley tree, whose
six-branch graphs are associated with dendritic polymers. According to Penner,
we construct the Teichm\"uller space of the decorated ideal octahedral surface
related to the quotient space of the fundamental group action. Using the
conformality of decoration, we define six moduli and the mapping class group
generated by cyclic permutations of the ideal vertices. Intending to quantize
the geometric area, we state the connection between the induced geometry and
the sine-Gordon model. Due to such a correspondence we obtain the differential
two-form in the cotangent bundle.
