Cannon and Thurston showed that a hyperbolic 3-manifold that fibers over the circle gives rise to a sphere filling curve. The universal cover of the fiber surface is quasi-isometric to the hyperbolic plane, whose boundary is a circle, and the universal cover of the 3-manifold is 3-dimension hyperbolic space, whose boundary is the 2-sphere. Cannon and Thurston showed that the inclusion map between the universal covers extends to a continuous map between their boundaries, whose image is dense. In particular, any measure on the circle pushes forward to a measure on the 2-sphere using this map. We consider Lebesgue measure on the circle, and the hitting measures associated to random walks on the surface group, and show that their push forwards onto the 2-sphere are mutually singular with the Lebesgue measure on the 2-sphere, and with the hitting measures from a random walk on the 3-manifold group.
This is joint work with Vaibhav Gadre, Thomas Haettel, Catherine Pfaff and Caglar Uyanik.