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Commentarii Mathematici Helvetici

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Volume 86, Issue 4, 2011, pp. 769–816
DOI: 10.4171/CMH/240

Published online: 2011-09-22

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

Christopher J. Leininger[1], Mahan Mj[2] and Saul Schleimer[3]

(1) University of Illinois at Urbana-Champaign, USA
(2) Tata Institute of Fundamental Research, Mumbai, India
(3) University of Warwick, Coventry, United Kingdom

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.

Keywords: Mapping class group, curve complex, ending lamination, Cannon--Thurston map

Leininger Christopher J., Mj Mahan, Schleimer Saul: The universal Cannon–Thurston map and the boundary of the curve complex. Comment. Math. Helv. 86 (2011), 769-816. doi: 10.4171/CMH/240