Non-arithmetic uniformization of metric spaces attached to unitary Shimura varieties

We develop a new method of constructing non-arithmetic lattices in the projective orthogonal group \(\text{PO}(n,1)\) for every integer \(n\) larger than one. The technique is to consider anti-holomorphic involutions on a complex arithmetic ball quotient, glue their fixed loci along geodesic subspaces, and show that the resulting metric space carries canonically the structure of a complete real hyperbolic orbifold. The volume of various of these non-arithmetic orbifolds can be explicitly calculated.