  Let $X$ be a regular curve and $n$ be a positive integer such that for every
nonempty open set $U\subset X$, there is a nonempty connected open set
$V\subset U$ 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.
