Rigidity and quasisymmetric uniformization of Thurston-type maps

TL;DR

This paper proves the No Invariant Line Fields conjecture and establishes a quasisymmetric uniformization theorem for a class of generalized postcritically-finite maps on higher-dimensional Riemannian manifolds, advancing understanding of their geometric structure.

Contribution

It introduces a new class of generalized postcritically-finite maps and proves key conjectures and uniformization results for them.

Findings

No Invariant Line Fields conjecture is proven for the class.

A quasisymmetric uniformization theorem is established.

The results extend to higher-dimensional Riemannian manifolds.

Abstract

We prove the No Invariant Line Fields conjecture for a class of generalized postcritically-finite branched covers on higher-dimensional Riemannian manifolds. Moreover, we establish a quasisymmetric uniformization theorem for this class of generalized postcritically-finite maps.