Sublinearly Morse Boundary II: Proper geodesic spaces

Abstract

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 $κ$, we construct a boundary for $X$, denoted $\mathcal{\partial}_κ X$, 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, then with minimal assumptions the Poisson boundary of $G$ can be realized on the $κ$-Morse boundary of $G$ equipped the word metric associated to any finite generating set.

Figures (10)

• Figure 1: $\Norm y = \Norm z$ and $q \in \pi_{\alpha} (z)$ as in the proof of Lemma \ref{['L:symmetry']}.
• Figure 2: Definition of $\kappa$-Morse set $Z$: Every quasi-geodesic ray $\beta$ has the property that there exists $R(Z, r, q, Q, \kappa')$, such that if $\beta_{R}$ is distance $\kappa'(R)$ from $Z$, then $\beta|_{r}$ is in the neighborhood $\mathcal{N}_{\kappa}(Z, m_{Z}(q, Q))$.
• Figure 3: The setup in the proof of Lemma \ref{['L:equivalence']} (i).
• Figure 4: Corollary \ref{['Cor:m_beta']}: $\Norm {z'_{r}} = r$ and $p = \pi_{\beta_{0}}(z'_{r})$ and $q = \pi_{\beta}(p)$.
• Figure 5: The proof of Claim \ref{['claim1']}.

Theorems & Definitions (97)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Remark 1.1
• Theorem F
• Lemma 2.1
• Lemma 2.2
• proof