  Sofic entropy is an invariant for probability-preserving actions of sofic
groups. It was introduced a few years ago by Lewis Bowen, and shown to extend
the classical Kolmogorov-Sinai entropy from the setting of amenable groups.
Some parts of Kolmogorov-Sinai entropy theory generalize to sofic entropy, but
in other respects this new invariant behaves less regularly.
  This paper explores conditions under which sofic entropy is additive for
Cartesian products of systems. It is always subadditive, but the reverse
inequality can fail. We define a new entropy-notion in terms of probability
distributions on the spaces of good models of an action. Using this, we prove a
general lower bound for the sofic entropy of a Cartesian product in terms of
separate quantities for the two factor systems involved. We also prove that
this lower bound is optimal in a certain sense, and use it to derive some
sufficient conditions for the strict additivity of sofic entropy itself.
Various other properties of this new entropy notion are also developed.
