Growth of nonuniform lattices in pinched, negatively curved manifolds

We study the relation between the entropy E(X) (exponential growth rate) of a Cartan-Hadamard manifold X of negative curvature and the critical exponent d(G) of its lattices G. When G is uniform, i.e. the quotient X/G is compact, the equality E(X)=d(G) is well-known; by some results of Eskin and McMullen, the same property holds true for non-uniform lattices (i.e. when the quotient X/G has finite volume) provided that X is symmetric. First, we give a direct geometrical proof of this fact, which holds for general 1/4-pinched (or also homogeneous) negatively curved spaces; then, we give examples showing that, surprisingly, the equality E(X)=d(G) fails for general negatively curved manifolds. We also show that the 1/4-pinching condition is optimal, as our examples have curvature arbitrarily close to [-4,-1].