A metric space \((X, d)\) is \(c\)-quasiconvex, \(c \geq 1\), if every pair of points \(x\), \(y\) in \(X\) can be joined by a rectifiable curve whose length is not more than \(c d(x, y)\). Let \(H = \mathbb R^3\) with coordinates \((x, y, t)\) be the Heisenberg group with Carnot-Carathéodory path metric (see, e.g., [R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1044.53022)]. The main result of the paper is that, if \(A\) is a closed subset of the Heisenberg group whose vertical projections into the \(xt\)- and \(yt\)-planes are nowhere dense, then the complement of \(A\) is quasiconvex.
As a corollary the authors prove that, if \(A \subset H\) is a closed set of cc-Hausdorff 3-measure zero, then \(H \setminus A\) is quasiconvex. In particular, sets of cc-Hausdorff dimension strictly less that \(3\) have quasiconvex complements, and another result of the paper shows that this is sharp: there exists a compact and totally disconnected set \(A \subset H\) of cc-Hausdorff dimension \(3\) whose complement is not quasiconvex.