Common finite covers and quasitrees

Sam Shepherd and I proved that if two finite simplicial complexes have isomorphic universal covers and free fundamental groups, then they have a common finite cover.
In the 1-dimensional case (graphs), this is Leighton's Theorem. I shall discuss various extensions of Leighton's Theorem and examples that illustrate its limitations, with
a focus on coverings by quasitrees. I shall then outline a proof of our theorem, which involves the construction of auxiliary CAT(0) cube complexes.