Sublinearly Morse boundary, II: Proper geodesic spaces

Qing, Yulan¹ ; Rafi, Kasra² ; Tiozzo, Giulio²

¹ Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China
² Department of Mathematics, University of Toronto, Toronto, ON, Canada

Geometry & topology, Tome 28 (2024) no. 4, pp. 1829-1889.

Résumé

We build an analogue of the Gromov boundary for any proper geodesic metric space, hence for any finitely generated group. More precisely, for any proper geodesic metric space $X$ and any sublinear function $\kappa$, we construct a boundary for $X$, denoted by ${\partial }_{\kappa }X$, that is quasi-isometrically invariant and metrizable. As an application, we show that when $G$ is the mapping class group of a finite type surface or a relatively hyperbolic group, with minimal assumptions, the Poisson boundary of $G$ can be realized on the $\kappa$–Morse boundary of $G$ equipped with the word metric associated to any finite generating set.

Détail

DOI : 10.2140/gt.2024.28.1829

Keywords: Morse geodesic, Poisson boundary, mapping class group, relatively hyperbolic group, geometric group theory, random walk

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