Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

Abstract

We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that every coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups -including mapping class groups of surfaces -are coarsely injective and coarsely injective groups are strongly shortcut.Using these results, we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, they have solvable conjugacy problem and finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.20F65, 20F67, 51F30

Stable cubulations, bicombings, and barycenters

“…In work that appeared simultaneously to ours, Haettel, Hoda and Petyt [35] proved that HHSs are coarse Helly spaces, in the sense of Chalopin, Chepoi, Genevois, Hirai and Osajda [20]. This property has a number of strong consequences, many of which overlap with the results in this paper.…”

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

“…A key ingredient of our proof for Theorem 1.1 is that the (extended) mapping class groups of orientable hyperbolic surfaces are semihyperbolic. Lemma 2.4 ([6, Corollary D], [8,Corollary 3.11]). For any orientable hyperbolic surface S of finite type, the mapping class group Mod.S / and the extended mapping class group Mod ˙.S / of S are semihyperbolic.…”

“…The mapping class group of a nonorientable surface N b g;p embeds in the mapping class group of the orientation double cover S 2b g 1;2p as the subgroup consisting of mapping classes that commute with the action of the deck group (see Lemma 2.7). In this paper, we prove Theorems 1.1 and 1.2 below by using the semihyperbolicity of the mapping class group of orientable surfaces, independently established by Durham-Minsky-Sisto [6, Corollary D] and Haettel-Hoda-Petyt [8,Corollary 3.11].…”