  We study the topological complexities of relative entropy zero extensions
acted by countableinfinite amenable groups. Firstly, for a given Folner
sequence $\{F_n\}_{n=0}^\infty$, we define respectively the relative entropy
dimensions and the dimensions of the relative entropy generating sets to
characterize the sub-exponential growth of the relative topological complexity.
Meanwhile, we investigate the relations among them. Secondly, we introduce the
notion of a relative dimension set. Moreover, using it, we discuss the
disjointness between the relative entropy zero extensions which generalizes the
results of Dou, Huang and Park[Trans. Amer. Math. Soc. 363(2) (2011), 659-680].
