Hausdorff dimension of limit sets

Abstract

We exhibit a class of Schottky subgroups of $\mathbf{PU}(1,n)$ ($n\ge 2$) 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.