Equivariant, almost homeomorphic maps between $S^ 1$ and $S^ 2$

Abstract:

Let $\Pi$ be a Fuchsian group isomorphic to a non-trivial, closed surface group, and let $M = {\mathbb {H}^3}/\Gamma$ be a hyperbolic 3-manifold admitting an isomorphism $\rho :\Pi \to \Gamma$. Under certain assumptions, Cannon-Thurston and Minsky showed that there exists a $\rho$-equivariant, surjective, continuous map $f:S_\infty ^1 \to S_\infty ^2$. In this paper, we prove that there exist zero-measure sets ${\Lambda ^1}$ in $S_\infty ^1$ and ${\Lambda ^2}$ in $S_\infty ^2$ such that the restriction $f{|_{S_\infty ^1 - {\Lambda ^1}}}:S_\infty ^1 - {\Lambda ^1} \to S_\infty ^2 - {\Lambda ^2}$ is a homeomorphism.