The quasisymmetric, quasiconformal, and quasi-isometric classification of geometrically finite Kleinian group limit sets is governed by the absence of Sierpiński carpet subsets and by the homogeneity of rank-two cut points.
On quasiconformal equivalence of Schottky regions
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abstract
In a recent paper, H. Shiga proved that the regions of discontinuity of any two Schottky groups of ranks at least two are quasiconformally equivalent. In this paper, we provide an alternative proof of such a fact. Our approach permits us to discuss quasiconformality equivalence of regions of discontinuity of Schottky type groups in terms of their signatures.
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Quasisymmetric universality, quasi-isometric classification and topological rigidity of Kleinian groups
The quasisymmetric, quasiconformal, and quasi-isometric classification of geometrically finite Kleinian group limit sets is governed by the absence of Sierpiński carpet subsets and by the homogeneity of rank-two cut points.