Journal of the European Mathematical Society


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Volume 17, Issue 4, 2015, pp. 755–762
DOI: 10.4171/JEMS/517

1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs

Sebastian Hensel[1], Piotr Przytycki[2] and Richard C.H. Webb[3]

(1) Department of Mathematics, University of Chicago, 5734 S. University Avenue, IL 60637, Chicago, USA
(2) Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke St West, QC H3A 0B9, Montreal, Canada
(3) Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, UK

We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

Keywords: Gromov hyperbolic, slim triangle, curve graph, arc graph, unicorn

Hensel Sebastian, Przytycki Piotr, Webb Richard: 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs. J. Eur. Math. Soc. 17 (2015), 755-762. doi: 10.4171/JEMS/517