  We relate Fuglede-Kadison determinants to entropy of algebraic actions of
sofic groups in essentially complete generality. This generalizes recent
results of Hanfeng Li and Andreas Thom from the amenable case to the sofic
case, as well as results of David Kerr, Hanfeng Li, and Lewis Bowen in the
residually finite case. Moreover, the proof given is the first calculation of
entropy of algebraic actions to avoid a nontrivial determinant approximation.
We apply our results to a problem of Christopher Deninger, generalizing results
of Hanfeng Li and David Kerr. Finally, we show that in many cases, finiteness
of topological entropy for algebraic actions of sofic groups is equivalent to
having metric mean dimension zero.
