Measure singularity for Cannon-Thurston maps

Vaibhav Gadre (University of Glasgow)

Monday 30th September 16:00-17:00 Maths 311B

Abstract

A hyperbolic 3-manifold that fibers over the circle gives an inclusion of the universal cover of the fibre, thought of as the hyperbolic plane, into the universal cover of the 3-manifold, namely hyperbolic 3-space. The inclusion has exponential distortion for the hyperbolic metrics on domain and the target. Despite the distortion, Cannon-Thurston showed that there is a well defined continuous map at infinity from the circle to the two-sphere. It follows that the image of the circle is a space filling curve. In present work with Maher, Pfaff and Uyanik, we show that the pushforward of many natural measures on the circle by the Cannon-Thurston map is singular with respect to similar natural measures on the two-sphere.

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