  We show that if $A$ is a closed subset of the Heisenberg group whose vertical
projections are nowhere dense, then the complement of $A$ is quasiconvex. In
particular, closed sets which are null sets for the cc-Hausdorff $3$-measure
have quasiconvex complements. Conversely, we exhibit a compact totally
disconnected set of Hausdorff dimension three whose complement is not
quasiconvex.
