Quasiconvexity in the Heisenberg group

TL;DR

This paper investigates the quasiconvexity of complements of certain closed sets in the Heisenberg group, establishing conditions under which these complements are quasiconvex and providing counterexamples.

Contribution

It characterizes when the complement of a closed set in the Heisenberg group is quasiconvex, linking it to properties like vertical projections and Hausdorff measure.

Findings

Complement of sets with nowhere dense vertical projections are quasiconvex.

Null sets for the cc-Hausdorff 3-measure have quasiconvex complements.

Existence of a compact totally disconnected set with non-quasiconvex complement.

Abstract

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.