Mappings of finite distortion: compactness of the branch set

We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, has a branch set homeomorphic to $(n-2)$-dimensional torus and distortion arbitrarily close to the asymptotic bound.