Number variance for homogeneous determinantal processes on hyperbolic spaces

We consider an abstract determinantal point process on a general non--elementary Gromov hyperbolic metric space governed by an orthogonal projection in the case when the space is homogeneous and the point process is invariant under isometries. We give a lower bound of the variance of the number of points inside a ball that is proportional to the volume of the ball. In particular, such point processes are never hyperuniform. Our result applies to the known examples of radial determinantal point processes on Cayley trees and on the standard hyperbolic spaces governed by Bergman projection kernels.