  In an attempt to understand the Tsallis entropy composition property, we
construct an embedding of the reals into the set of $3\times 3$ upper
triangular matrices with real entries. We explore consequences of this
embedding and of the geometry of the ambient $3\times 3$ Heisenberg group. This
approach establishes the polynomial growth of the volume of phase space of
systems described by the Tsallis entropy and provides a general framework for
understanding Abe's formula in terms of the Pansu derivative between Riemannian
spaces.
