# 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. AMS subject classification = 20F67(Primary), 22E40 57M50

### Cite this paper

@inproceedings{Leininger2009TheUC, title={The universal Cannon–Thurston map and the boundary of the curve complex}, author={Christopher J. Leininger and Mahan Mjand and Saul Schleimer}, year={2009} }