Sublinear bilipschitz equivalence and sublinearly Morse boundaries (Q6591568)

Sublinear bilipschitz equivalence (SBE) is an equivalence relation between metric spaces that generalizes quasi-isometry. It was first used in the work of \textit{Y. de Cornulier} [J. Topol. 1, No. 2, 342--361 (2008; Zbl 1157.22003)] as a means to compute dimensions of asymptotic cones. Since the Gromov-hyperbolicity admits a characterization in terms of asymptotic cones, it follows that being hyperbolic is an SBE-invariant property among compactly generated locally compact groups (see [\textit{Y. de Cornulier}, Ann. Sci. Éc. Norm. Supér. (4) 52, No. 5, 1201--1242 (2019; Zbl 1470.22001), Theorem 4.3]).\N\NIn this paper, the authors extend this result beyond groups and spaces that are Gromov-hyperbolic. More precisely, given a proper geodesic space that is not Gromov-hyperbolic, but exhibiting some features of Gromov-hyperbolic spaces, they study its large scale hyperbolic-like structure by describing the sublinearly Morse boundaries of the group. Furthermore, they show that the set of sublinearly Morse directions is invariant under suitable SBE. As an application, they distinguish a pair of right-angled Coxeter groups brought up by \textit{J. Behrstock} [Lond. Math. Soc. Lect. Note Ser. 454, 151--159 (2019; Zbl 1514.20137)] up to SBE. They also show that, under mild assumptions, generic random walks on countable groups are sublinear rays.