Gaps in the exponent spectrum of subgroups of discrete quasiconformal groups

Abstract

Let G be a discrete quasiconformal group preserving B³ whose limit set Λ(G) is purely conical and all of ∂B³. Let Ĝ be a non-elementary normal subgroup of G: we show that there exists a set $\mathscr{A}$ of full measure in Λ(G) so that $\mathscr{A}$, regarded as a subset of Λ (Ĝ), has "fat horospherical" dynamics relative to Ĝ. As an application we will bound from below the exponent of convergence of Ĝ in terms of the Hausdorff dimension of $\mathscr{A}$.