**MSC:**- 20H10 Fuchsian groups and their generalizations, See also {11F06, 22E40, 30F35, 32Nxx}
- 58C40 Spectral theory; eigenvalue problems, See also {47H12,

that for convex cocompact Kleinian groups

the exponent of convergence is bounded from above by an expression

which depends mainly on the diameter of the convex core of the associated

infinite-volume hyperbolic manifold.

This result is derived via

refinements of Sullivan's shadow lemma and of estimates for

the growth of the orbital

counting function and Poincare series. We

finally obtain

spectral and fractal implications, such as lower bounds for

the bottom of the spectrum

of the Laplacian on these manifolds, and upper bounds

for the decay of the area of neighbourhoods of the

associated limit sets.