From Hierarchical to Relative Hyperbolicity

Abstract

We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil–Petersson metric on Teichmüller space for surfaces with complexity three.

Largest acylindrical actions and Stability in hierarchically hyperbolic groups

“…In light of Theorem A.1 in the Appendix, this property is no longer required for this paper. We keep the contents of this section in the paper nonetheless, since they have found independent interest and already been used elsewhere, e.g., [BR,HS16,Rus20], as well as in several papers in progress.…”

Thickness and relative hyperbolicity for graphs of multicurves

“…Theorem 3.2 (Special case of [21,Theorem 4.3]). Let 𝔖 be the set of witnesses for some hierarchical graph of multicurves and let 𝔗 be the set of all (not necessarily connected) subsurfaces of 𝑆 where each component is an element of 𝔖.…”

“…Sultan showed that $$\operatorname{Sep}(S_{2,0})$$ is also hyperbolic [24]. Previous work of the authors completed the classification, showing that $$\operatorname{Sep}(S)$$ is hyperbolic if $$p \geqslant 3$$ or $$(g,p) \in \lbrace (2,0),(2,1),(1,2)\rbrace$$ [26], relatively hyperbolic if $$p =0$$ and $$g \geqslant 3$$ or $$p=2$$ and $$g \geqslant 2$$ [21], and thick of order at most 2 if $$p=1$$ and $$g \geqslant 3$$ [22]. These results again match up with the classification in terms of witnesses given in Theorem 2.25 and employ special cases of the techniques employed here.…”

“…Since $$\operatorname{Wit}\bigl (\operatorname{Sep}(S)\bigr ) = \operatorname{Wit}\bigl (\mathcal {T}(S)\bigr )$$, Proposition 2.14 implies that the inclusion of $$\operatorname{Sep}(S)$$ into $$\mathcal {T}(S)$$ is a quasi‐isometry whenever $$\mathcal {T}(S)$$ is connected. The statement on relative hyperbolicity follows from the relative hyperbolicity of $$\operatorname{Sep}(S)$$ when $$S$$ is closed; see [21]. $$\Box$$…”

“…Sultan showed that Sep(𝑆 2,0 ) is also hyperbolic [24]. Previous work of the authors completed the classification, showing that Sep(𝑆) is hyperbolic if 𝑝 ⩾ 3 or (g, 𝑝) ∈ {(2, 0), (2, 1), (1, 2)} [26], relatively hyperbolic if 𝑝 = 0 and g ⩾ 3 or 𝑝 = 2 and g ⩾ 2 [21], and thick of order at most 2 if 𝑝 = 1 and g ⩾ 3 [22]. These results again match up with the classification in terms of witnesses given in Theorem 2.25 and employ special cases of the techniques employed here.…”