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

Abstract

For a pinched Hadamard manifold X and a discrete group of isometries Γ of X, the critical exponent δΓ 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 Γ0 has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients Γ0/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.