Sequence entropy and independence in free and minimal actions

For every countable infinite group that admits \mathbb{Z} as a homomorphic image, we show that for each m\in{\mathbb{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 {\mathbb{Z}}^{r} , for every m\in{\mathbb{N}} , we found a free minimal action with topological sequence entropy \log(n) , where m\leq n\leq m^{2^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\geq m .