# A Metrizable Topology on the Contracting Boundary of a Group

## TL;DR

This paper introduces a new metrizable topology on the contracting boundary of proper geodesic metric spaces, providing a quasi-isometry invariant that enhances understanding of the boundary's geometric structure.

## Contribution

The authors define a new, geometrically meaningful topology on the contracting boundary and prove its metrizability for Cayley graphs of finitely generated groups.

## Key findings

- The topology is invariant under quasi-isometries.
- The topology is metrizable for Cayley graphs of finitely generated groups.
- Provides a new framework for studying boundaries in geometric group theory.

## Abstract

The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.