Sequence entropy and independence in free and minimal actions

For every countable infinite group that admits Z as a homomorphic image, we show that for each m∈{N} , there exists a minimal action whose topological sequence entropy is log(m) . Furthermore, for every countable infinite group G that contains a finite index normal subgroup G' isomorphic to {Z}ʳ , for every m∈{N} , we found a free minimal action with topological sequence entropy log(n) , where m≤ n≤ m2^r[G:G'] . If, instead, G is assumed to be an infinite countable group that is a direct sum of finite groups, the previous result holds with the topological sequence entropy being equal to log(m) . In each of the above cases, we also show that the aforementioned minimal actions admit non-trivial independence tuples of size n but do not admit non-trivial independence tuples of size n+1 for some n≥ m .