We prove an inequality between the relative homological dimension of a Kleinian group Γ ⊂ Isom(Hn) and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere.

Abstract This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence of interface modes that are induced by properties bulk structure. For general 1D structure with time-reversal symmetry, we investigate an mode Dirac point upon perturbation. Specifically, establish conditions perturbation which guarantee opening band gap around a...