Uniform hyperbolicity of the curve graph via surgery sequences

Abstract

Abstract. We prove that the curve graph C(1) (S) is Gromovhyperbolic with a constant of hyperbolicity independent of the surface S. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.

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

“…Bowditch [Bow06] proved that k can be chosen to grow logarithmically with the complexity of S. A different proof of hyperbolicity was given by Hamenstädt [Ham07]. Recently, Aougab [Aou13], Bowditch [Bow14], and Clay, Rafi and Schleimer [CRS14] have proved, independently, that k can be chosen independent of S.…”

Stable commutator length on mapping class groups

“…-Let Σ h be the closed surface of genus h. For G = M CG(Σ g ) it would be interesting to know how s h := inf{scl(g) | g ∈ G, scl(g) > 0} behaves when the genus h → ∞. On the plus side, subsurface projection constants are uniform (see Leininger's proof in [29,30]) and so is the hyperbolicity constant δ of curve graphs, see [10,25,18]. The acylindicity constants in Lemma 2.4 are known explicitly [35], translation lengths in curve complexes are not uniform, but the asymptotics is understood [24].…”

Geometric approaches to braid groups and mapping class groups

“…Recently, several authors independently [1,6,11,22] proved something even more surprising: The constant δ which bounds the thinness of triangles in C 1 (S) is independent of the surface S! That is, thinness of triangles is uniformly bounded, regardless of the genus or the number of boundary components of S. Moreover, one can take a small number like δ = 17 [22].…”