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.
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Christopher J. Leininger, Mahan Mj, Saul Schleimer, The universal Cannon–Thurston map and the boundary of the curve complex. Comment. Math. Helv. 86 (2011), no. 4, pp. 769–816DOI 10.4171/CMH/240