Cannon--Thurston maps for Anosov foliations

TL;DR

This paper constructs a new type of Cannon--Thurston map for Anosov foliations with branching on hyperbolic manifolds, revealing complex group actions on universal circles.

Contribution

It introduces a novel Cannon--Thurston-type map for universal circles associated with Anosov foliations with branching.

Findings

The leftmost universal circle admits a Cannon--Thurston-type map to the ideal 2-sphere.

The fundamental group acts on this circle with pseudo-Anosov dynamics.

This provides new insights into the structure of universal circles in hyperbolic geometry.

Abstract

Universal circles, introduced by Thurston and Calegari--Dunfield, are not well understood in general. Recently, the author together with Taylor showed that Anosov foliations with branching admit nonconjugate universal circles. We continue the study of these universal circles and show that for an Anosov foliation with branching on a hyperbolic manifold, the leftmost universal circle admits a Cannon--Thurston-type map to the ideal 2-sphere. This is a new type of construction of a Cannon--Thurston map. As a corollary, we show the fundamental group of the manifold acts on the leftmost universal circle with pseudo-Anosov dynamics.