Sequence entropy and independence in free and minimal actions

Abstract

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 $G^{\prime }$ isomorphic to ${\mathbb{Z}}^r$, for every $m\in \mathbb{N}$, we found a free minimal action with topological sequence entropy $\log (n)$, where $m\le n\le m^{2^r[G:G^{\prime }]}$. 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 $n\ge m$.