The 4-dimensional surgery and 5-dimensional \(s\)-cobordism theorems are known to be true in the topological category for a certain class of groups called good groups. The class of previously known good groups coincides with the elementary amenable groups [the first author, Proc. Int. Congr. Math., Warszawa 1983, Vol. I, 647-663 (1984; Zbl 0577.57003)], i.e. the class of groups containing all groups of polynomial growth and closed under (1) extension and (2) direct limit. In this fundamental paper, the authors expand the class of known good groups to contain all groups of subexponential growth and those that can be obtained from these by a finite number of application of operations (1) and (2). This is an effective expansion because it is known that elementary amenable groups grow either polynomially or exponentially, and that there are (uncountably many) groups of intermediate growth [\textit{R. I. Grigorchuk}, Math. USSR, Izv. 25, 259-300 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 939-985 (1984; Zbl 0583.20023)], i.e. groups that grow faster than any polynomial but slower than any exponential function.