Uniform Hyperbolicity of the Graphs of Curves

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

“…A different proof of hyperbolicity was given by Hamenstädt [Ham07]. Recently, Aougab [Aou12], Bowditch [Bow12], and Clay, Rafi and Schleimer [CRS13] have proved, independently, that k can be chosen independent of S.…”

Uniform hyperbolicity of the curve graph via surgery sequences

“…We also prove uniform hyperbolicity in the closed case, when n = 0. This follows from Theorem 6.4, as C (1) (S g,0 ) isometrically embeds in C (1) (S g,1 ).…”

“…Note that these are flag complexes: when the one-skeleton of a simplex is present, so is the simplex itself. Let K (1) denote the one-skeleton of a simplicial complex K.…”

“…The inclusion C (1) (S g,n ) → AC (1) (S g,n ) gives a quasi-isometric embedding with constants independent of g and n. Deduce the following.…”

“…In all of these the upper bound on the hyperbolicity constant depended on the topology of the surface S. During the process of writing this paper, several other proofs of uniform hyperbolicity emerged. Bowditch [5] has refined his approach to obtain uniform constants using techniques he developed in [4]; the proof by Aougab [1] has many common themes with the work of Bowditch. The work of Hensel, Przytycki, and Webb [10] also uses surgery paths and has other points of contact with our work.…”