The quasi-redirecting Boundary

Abstract

We generalize the notion of Gromov boundary to a larger class of metric spaces beyond Gromov hyperbolic spaces. Points in this boundary are classes of quasi-geodesic rays and the space is equipped with a topology that is naturally invariant under quasi-isometries. It turns out that this boundary is compatible with other notions of boundary in many ways; it contains the sublinearly Morse boundary as a topological subspace and it matches the Bowditch boundary of relative hyperbolic spaces when the peripheral subgroups have no intrinsic hyperbolicity. We also give a complete description of the boundary of the Croke-Kleiner group where the quasi-redirecting boundary reveals a new class of QI-invariant, Morse-like quasi-geodesics.

Figures (9)

• Figure 2: The ray $\alpha$ can be quasi-redirected to $\beta$ at radius $r$.
• Figure 3: The ray $\zeta$, which is constructed from ray $\zeta_1$ and $\zeta_2$, quasi-redirects $\alpha$ to $\gamma$.
• Figure 4: The space $X_k$ is a proper geodesic metric space. However, $[\beta]$ does not have a geodesic representative.
• Figure 5: Weakly Morse: The $(\kappa, n)$--neighbourhood of the geodesic ray $b$.
• Figure 6: The path $[{\mathfrak o}, q] \cup P \cup \beta_{[z, \infty)}$ is a quasi-geodesic ray that redirects $\alpha$ to $\beta$.

Theorems & Definitions (137)

• Proposition A
• Theorem B
• Theorem C
• Theorem E
• Theorem F
• Corollary G
• Theorem I
• Definition \oldthetheorem: Quasi Isometric embedding
• Definition \oldthetheorem: Quasi-Geodesics
• Lemma \oldthetheorem: Taming of the quasi-geodesics