Moebius and sub-Moebius structures

TL;DR

This paper introduces sub-Moebius structures, establishes conditions for them to be Moebius structures, and shows that the boundary at infinity of Gromov hyperbolic spaces naturally carries a canonical sub-Moebius structure compatible with the standard topology.

Contribution

It defines sub-Moebius structures, provides criteria for their equivalence to Moebius structures, and demonstrates their natural occurrence on Gromov hyperbolic space boundaries.

Findings

Necessary and sufficient conditions for sub-Moebius to Moebius structures.

Existence of a canonical sub-Moebius structure on Gromov hyperbolic boundaries.

Invariance of the sub-Moebius structure under isometries.

Abstract

We introduce a notion of a sub-Moebius structure and find necessary and sufficient conditions under which a sub-Moebius structure is a Moebius structure. We show that on the boundary at infinity of every Gromov hyperbolic space Y there is a canonical sub-Moebius structure which is invariant under isometries of Y such that the sub-Moebius topology on the boundary coincides with the standard one.