# Commentarii Mathematici Helvetici

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**Volume 86, Issue 4, 2011, pp. 769–816**

**DOI: 10.4171/CMH/240**

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

Christopher J. Leininger^{[1]}, Mahan Mj

^{[2]}and Saul Schleimer

^{[3]}(1) Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green Street, IL 61801, URBANA, UNITED STATES

(2) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, 400005, MUMBAI, INDIA

(3) Department of Mathematics, University of Warwick, CV4 7AL, 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, 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