Action rigidity for free products of hyperbolic manifold groups

Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry rank-1 symmetric spaces prove, within each quasi-isometry class, that residually finite are abstractly commensurable. Our result gives first examples hyperbolic quasi-isometric but do not virtually geometry. An important component proof is generalization Leighton’s graph covering theorem. The main theorem depends residual finiteness, we show extensions would give counterexamples.