Sensitive group actions on regular curves of almost $leq n$ order

Let $X$ be a regular curve and $n$ be a positive integer such that for every nonempty open set $Usubset X$, there is a nonempty connected open set $Vsubset U$ with the cardinality $|partial_X(V)|leq n$. We show that if $X$ admits a sensitive action of a group $G$, then $G$ contains a free subsemigroup and the action has positive geometric entropy. As a corollary, $X$ admits no sensitive nilpotent group action.