  We investigate the possibility of discrete groups furnishing a kinematic
framework for systems whose thermodynamic behaviour may be given by
non-additive entropies. Relying on the well-known result of the growth rate of
balls of nilpotent groups, we see that maintaining extensivity of the entropy
of a nilpotent group requires using a non-Boltzmann/Gibbs/Shannon (BGS)
entropic form. We use the Tsallis entropy as an indicative alternative. Using
basic results from hyperbolic and random groups, we investigate the genericity
and possible range of applicability of the BGS entropy in this context. We
propose a sufficient condition for phase transitions, in the context of
(multi-) parameter families of non-additive entropies.
