Reconstruction of Anosov flows from infinity

Abstract

Every pseudo-Anosov flow $\varphi$ in a closed $3$-manifold $M$ gives rise to an action of ${\pi }_1(M)$ on a circle $S_{\infty }^1(\varphi )$ from infinity \cite{Fen12}, with a pair of invariant \emph{almost} laminations. From certain actions on $S^1$ with invariant almost laminations, we reconstruct flows and manifolds realizing these actions, including all orientable transitive pseudo-Anosov flows in closed $3$-manifolds. Our construction provides a geometry model for such flows and manifolds induced from $\mathcal{D}\times \mathcal{D}$, where $\mathcal{D}$ is the Poincaré disk with $\partial \mathcal{D}$ identified with $S_{\infty }^1(\varphi )$. In addition, our result applies to Cannon conjecture under the assumption that certain group-equivariant sphere-filling Peano curve exists, which offers a description of orientable quasigeodesic pseudo-Anosov flows in hyperbolic $3$-manifolds in terms of group actions on $\partial {\mathbb{H}}^3\times \partial {\mathbb{H}}^3\times \partial {\mathbb{H}}^3$.

arXiv comment62 pages, 19 figures. Comments welcome!