Hausdorffness under bounded leaf volumes and singular-set dimension assumptions

Determine whether uniform boundedness of the volumes of all regular leaves and the inequality that the complex dimension of the singular set is strictly less than the leaf dimension alone imply that the quotient by the equivalence relation generated by intersecting leaf closures is Hausdorff.

Background

The paper defines an equivalence relation by joining points through finite chains of intersecting closures of regular leaves. The main global quotient theorem proves that this relation is closed, and hence that the quotient is compact Hausdorff, under three hypotheses: a uniform bound on regular-leaf volumes, a singular set of dimension strictly smaller than the leaf dimension, and finite Hausdorff measure of the union of leaf closures meeting the singular set.

Proposition 1.4 shows that the finite-measure assumption can be replaced by the weaker condition that only finitely many equivalence classes meet the singular set and that each such class is closed. The authors explicitly note that finite measure is therefore sufficient but not necessary for Hausdorffness. They leave unresolved whether the first two hypotheses alone suffice; the paper's counterexample later demonstrates that a uniform leaf-volume bound alone does not imply closedness, but it does not settle the case when the singular-set dimension is strictly smaller than the leaf dimension.