Hausdorff dimension of limit sets

TL;DR

This paper investigates the Hausdorff dimension of limit sets for certain Schottky subgroups of complex hyperbolic space, establishing equality with the growth exponent for well-positioned groups and providing bounds for general cases.

Contribution

It introduces the concept of well-positioned Schottky subgroups and proves the Hausdorff dimension equals the growth exponent for these groups, extending understanding of limit set dimensions.

Findings

Hausdorff dimension equals growth exponent for well-positioned groups

Provides lower bounds for general groups involving Patterson-Sullivan measures

Utilizes Ledrappier-Young theorem as a key analytical tool

Abstract

We exhibit a class of Schottky subgroups of $\mathbf{PU}(1,n)$ ($n \geq 2$) which we call well-positioned and show that the Hausdorff dimension of the limit set $\Lambda_\Gamma$ associated with such a subgroup $\Gamma$, with respect to the spherical metric on the boundary of complex hyperbolic $n$-space, is equal to the growth exponent $\delta_\Gamma$. For general $\Gamma$ we establish (under rather mild hypotheses) a lower bound involving the dimension of the Patterson-Sullivan measure along boundaries of complex geodesics. Our main tool is a version of the celebrated Ledrappier-Young theorem.