The Poincaré exponent and the dimensions of Kleinian limit sets
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Kleinian groups are discrete groups of isometries of hyperbolic space. Their actions give rise to intricate fractal limit sets on the boundary at infinity and there is great interest in estimating the “dimension” of these limit sets. As an invitation to this fascinating area, we provide a proof of the (well-known) result that the Poincaré exponent of a nonelementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof uses only elementary hyperbolic and fractal geometry.
Fraser , J M 2022 , ' The Poincaré exponent and the dimensions of Kleinian limit sets ' , The American Mathematical Monthly , vol. 129 , no. 5 , pp. 480-484 . https://doi.org/10.1080/00029890.2022.2041362
The American Mathematical Monthly
Copyright © 2022 The Mathematical Association of America. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1080/00029890.2022.2041362.
DescriptionThe author was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034).
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