The universal Cannon-Thurston map and the boundary of the curve complex

Abstract

In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent-Leininger-Schleimer and Mitra, we construct a universal Cannon-Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected. © 2011 Christopher J. Leininger, Mahan Mj and Saul Schleimer.