Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distance

For a pinched Hadamard manifold X and a discrete group of isometries Γ of X, the critical exponent δ _Γ is the exponential growth rate of the orbit of a point in X under the action of Γ. We show that the critical exponent for any family N of normal subgroups of Γ ₀ has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients Γ ₀ /Γ. The key tool is to analyse the spectrum of transfer operators associated to subshifts of finite type, for which we obtain a result of independent interest.