The Gromov boundary of the ray graph

TL;DR

This paper characterizes the Gromov boundary of the ray graph, a hyperbolic graph related to the plane minus a Cantor set, showing it is homeomorphic to a quotient of a circle, with implications for the mapping class group's action.

Contribution

It provides a detailed description of the Gromov boundary of the ray graph in terms of cliques of long rays, linking geometric boundary structure to topological quotient spaces.

Findings

Gromov boundary described via cliques of long rays

Homeomorphism to a quotient of a subset of the circle

Enhanced understanding of the mapping class group's action on the boundary

Abstract

The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray graph is homeomorphic to a quotient of a subset of the circle. This version contains some updates and corrections.