We　study　the　topological　complexities　of　relative　entropy　zero　extensions　acted　upon　by　countable-infinite　amenable　groups.　First,　for　a　given　Følner　sequence　{Fn}n=0+∞　,　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.　©　2023,　Innovation　Academy　for　Precision　Measurement　Science　and　Technology,　Chinese　Academy　of　Sciences.