Quotients of Torus Endomorphisms and Lattès-Type Maps

We show that if an expanding Thurston map is the quotient of a torus endomorphism, then it has a parabolic orbifold and is a Lattès-type map.

On Cannon–Thurston maps for relatively hyperbolic groups

Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon–Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon–Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.