  We present an embedding of the Tsallis entropy into the 3-dimensional
Heisenberg group, in order to understand the meaning of generalized
independence as encoded in the Tsallis entropy composition property. We infer
that the Tsallis entropy composition induces fractal properties on the
underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we
justify why the underlying configuration/phase space of systems described by
the Tsallis entropy has polynomial growth for both discrete and Riemannian
cases. We provide a geometric framework that elucidates Abe's formula for the
Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian
spaces.
