Hausdorff Dimension of non-conical and Myrberg limit sets

TL;DR

This paper investigates the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces, establishing maximality in various geometric contexts and confirming a conjecture relating Myrberg limit set dimension to the critical exponent.

Contribution

It introduces new techniques to analyze Hausdorff dimensions of limit sets and proves maximality results for non-conical limit sets across different negatively curved spaces.

Findings

Maximal Hausdorff dimension of non-conical limit sets in various geometric settings

Dimension of Myrberg limit set equals the critical exponent

Confirms Falk-Matsuzaki conjecture

Abstract

In this paper, we develop techniques to study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set of $G$ in the following cases. 1. $M$ is a finite volume complete Riemannian manifold of pinched negative curvature and $G$ is an infinite normal subgroups of infinite index in $\pi_1(M)$. 2. $G$ acts on a regular tree $X$ with $X/G$ infinite and amenable (dimension 1). 3. $G$ acts on the hyperbolic plane $\mathbb H^2$ such that $\mathbb H^2/G$ has Cheeger constant zero (dimension 2). 4. $G$ is a finitely generated geometrically infinite Kleinian group (dimension 3). We also show that the Hausdorff dimension of the Myrberg limit set is the same as the critical exponent, confirming a conjecture of Falk-Matsuzaki.