Dimension of limit sets in variable curvature

TL;DR

This paper calculates the Hausdorff dimension of limit sets for Kleinian groups acting on negatively curved manifolds and constructs hyperbolic surfaces with specific orbit properties, advancing understanding of geometric group actions.

Contribution

It provides a general formula for the Hausdorff dimension of limit sets in variable negative curvature and constructs examples with zero-dimensional non-recurrent orbits.

Findings

Hausdorff dimension formula for limit sets in variable curvature

Construction of hyperbolic surfaces with zero-dimensional non-recurrent orbits

Extension of dimension theory to broader class of Kleinian groups

Abstract

We compute the Hausdorff dimension of the limit set of an arbitrary Kleinian group of isometries of a complete simply-connected Riemannian manifold with pinched negative sectional curvatures $-b^2\leq k\leq -1$. Moreover, we construct hyperbolic surfaces with a set of non-recurrent orbits of dimension zero.