The Poincaré exponent and the dimensions of Kleinian limit sets

Jonathan M. Fraser

doi:10.1080/00029890.2022.2041362

The Poincaré exponent and the dimensions of Kleinian limit sets

Research output: Contribution to journal › Comment/debate › peer-review

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 language	English
Pages (from-to)	480-484
Number of pages	5
Journal	The American Mathematical Monthly
Volume	129
Issue number	5
Early online date	11 Mar 2022
DOIs	https://doi.org/10.1080/00029890.2022.2041362
Publication status	Published - 2022

Keywords

General Mathematics

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10.1080/00029890.2022.2041362

Fraser-2022-Poincaré-exponent-AMM-AAM
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.
Accepted author manuscript, 222 KB

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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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 Poincar{\'e} 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.",

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AU - Fraser, Jonathan M.

N1 - The author was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034).

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

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DO - 10.1080/00029890.2022.2041362

M3 - Comment/debate

SN - 0002-9890

VL - 129

SP - 480

EP - 484

JO - The American Mathematical Monthly

JF - The American Mathematical Monthly

IS - 5

ER -

The Poincaré exponent and the dimensions of Kleinian limit sets

Abstract

Keywords

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Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this