The classical theory of entropy of measure-preserving transformations of a group \(G\), or of continuous actions of a group \(G\) extended naturally from the original setting \(G=\mathbb{Z}\) to the case of \(G\) being a countable amenable group. In these developments fundamental inequalities survived, expressing the idea that the entropy of a system is at least as big as the entropy of any of its factor systems: roughly speaking, ‘bigger’ systems cannot have less entropy than ‘smaller’ ones. A particularly convenient class of examples which have a very straightforward relationship between (Haar) measure theoretic and topological entropy, are the actions by continuous automorphisms of a compact group. Here tools like the Yuzvinskii addition formula, showing that entropy adds over \(G\)-equivariant exact sequences of compact groups and continuous homomorphisms with closed image, give a more precise articulation of that monotonicity property of entropy. D. S. Ornstein and B. Weiss [J. Anal. Math. 48, 1–141 (1987; Zbl 0637.28015)], alongside their development of the Ornstein theory for actions of countable amenable groups, pointed out that the full shift with alphabet given by the group \(C_2\) with two elements over the free group \(G=F_2\), a natural example of an action of \(F_2\) by automorphisms of a compact group, factors onto the full shift with alphabet \(C_2\times C_2\). Whatever one might mean by entropy in this situation, the former system has entropy \(\log2\) while the latter factor system has entropy \(\log4\). This clear indication that entropy theory for actions of non-amenable groups could not satisfy monotonicity results about factors, nor addition formulas like the Yuzvinskii addition formula gave rise to the view that there would be no good entropy theory beyond the amenable context. New ideas were brought into the picture by L. Bowen [J. Am. Math. Soc. 23, No. 1, 217–245 (2010; Zbl 1201.37005)], later refined by D. Kerr and H. Li [Invent. Math. 186, No. 3, 501–558 (2011; Zbl 1417.37041)], which showed that there was indeed an entropy theory for sofic (and thus, possibly, all) countable groups, which in particular specialized to the familiar theory for amenable groups and that it does indeed - necessarily - have surprising new features.
In the present paper the factor map which increases the entropy for a free group action used to construct the example of Ornstein and Weiss is generalized, and the entropy increase is related to the concept of cost and to the first \(\ell^2\)-Betti numbers of the acting group. This relationship is also developed in the more general setting of coboundary maps arising from simplicial actions and the relationship between \(\ell^2\)-Betti numbers and the failure of the Yuzvinskii addition formula is explored. In order to do this, other more fundamental properties of the entropy theory of algebraic actions of profinite groups are developed. In particular, for this setting it is shown that the topological sofic entropy coincides with the Haar measure-theoretic sofic entropy whenever the homoclinic subgroup of the action is dense.