Bernd O. Stratmann
A Note on Geometric Upper Bounds for the Exponent of Convergence of Convex Cocompact Kleinian Groups
Preprint series: Mathematica Gottingensis

MSC:

20H10 Fuchsian groups and their generalizations

58C40 Spectral theory; eigenvalue problems

ZDM: 11F72
CR: 20E40
Abstract: In this note we obtain by purely geometric means

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.


Keywords: Kleinian groups, hyperbolic geometry, fractal geometry, spectral theory