Universal Cannon-Thurston maps and the boundary of the curve complex

Abstract

The fundamental group of a closed surface of genus at least two admits a natural action 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. Further, we show that the boundary of this curve complex is locally path-connected.