The Poincaré exponent and the dimensions of Kleinian limit sets

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Abstract

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.
Original languageEnglish
Pages (from-to)480-484
Number of pages5
JournalThe American Mathematical Monthly
Volume129
Issue number5
Early online date11 Mar 2022
DOIs
Publication statusPublished - 2022

Keywords

  • General Mathematics

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