Remarks on Hausdorff dimensions for transient limits sets of Kleinian groups
Let \(G\) be a Kleinian group acting on \(N+1\)-dimensional hyperbolic space. The authors call \(G\) a ``discrepency group'' if the Hausdorff dimension of the limit set of \(G\) strictly exceeds its critical exponent. They show that it is possible to fractionate the limit set into a family of subsets of increasing Hausdorff dimension. This is achieved by an upper bound derived from a covering argument. The authors also prove that if \(H\) is a non-elementary Kleinian group and if \(G\) is a non-trivial normal subgroup of \(H\) then the critical exponent of \(G\) is at least the half of that of \(H\). They derive this unexpected result from an observation of \textit{K. Matsuzaki} [Comp. Methods Funct. Theory 2, 469--479 (2002; Zbl 1062.30052)]. They conclude the paper with a discussion of a number of examples of deficiency groups.
scientific article; zbMATH DE number 2157245
scientific article