A general collapsing result for families of stratified Riemannian metrics on orbifolds

This paper proves a general collapsing result for families of stratified Riemannian metrics ${\widehat{g}}^{\mu }$ on a compact orbifold $E$, subject to suitable limiting conditions on the metrics ${\widehat{g}}^{\mu }$ as $\mu \to \mathrm{\infty }$. The result is distinct from similar theorems in the literature since it does not require bounds on curvature or injectivity radius of $(E,{\widehat{g}}^{\mu })$ and thus allows for Gromov-Hausdorff limits of $(E,{\widehat{g}}^{\mu })$ which have strictly lower dimension than $E$. The paper also introduces and studies a new class of stratified fibrations between orbifolds, termed weak submersions, and new classes of geometric structures on orbifolds, termed stratified Riemannian metrics, stratified Riemannian semi-metrics and stratified quasi-Finslerian structures, all of which play a key role in the proof of the main theorem.