JournalscmhVol. 86, No. 4pp. 769–816

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

  • Christopher J. Leininger

    University of Illinois at Urbana-Champaign, USA
  • Mahan Mj

    Tata Institute of Fundamental Research, Mumbai, India
  • Saul Schleimer

    University of Warwick, Coventry, United Kingdom
The universal Cannon–Thurston map and the boundary of the   curve complex cover
Download PDF

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.

Cite this article

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. 769–816

DOI 10.4171/CMH/240