Abstract: We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups. First, for a given Følner sequence $\{F_n\}_{n=0}^{+\infty}$, we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. we also investigate the relations among these. Second, we introduce the notion of a relative dimension set. Moreover, using the method, we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions, which says that if the relative dimension sets of two extensions are different, then the extensions are disjoint.

Key words: amenable groups, relative entropy dimensions, relative dimension sets