Equivalent Topologies on the Contracting Boundary

TL;DR

This paper proves that the contracting boundary and the 1-Morse boundary are topologically equivalent in proper geodesic metric spaces, confirming a conjecture about their relationship.

Contribution

It establishes the equivalence of the contracting boundary and the 1-Morse boundary, clarifying their relationship in the context of hyperbolic space generalizations.

Findings

Contracting boundary is a topological space generalizing Gromov boundary.

The 1-Morse boundary and contracting boundary are equivalent when ppa=1.

Confirms the conjecture relating these two boundary concepts.

Abstract

The contracting boundary of a proper geodesic metric space generalizes the Gromov boundary of a hyperbolic space. It consists of contracting geodesics up to bounded Hausdorff distances. Another generalization of the Gromov boundary is the $\kappa$-Morse boundary with a sublinear function $\kappa$. The two generalizations model the Gromov boundary based on different characteristics of geodesics in Gromov hyperbolic spaces. It was suspected that the $\kappa$-Morse boundary contains the contracting boundary. We will prove this conjecture: when $\kappa =1$ is the constant function, the 1-Morse boundary and the contracting boundary are equivalent as topological spaces.