There are no geodesic hubs in the Brownian sphere

TL;DR

This paper proves that in the Brownian sphere, no point serves as a hub for three or more disjoint geodesics, revealing a unique geometric property of this random metric space.

Contribution

It establishes the non-existence of $k$-hubs for $k eq 2$ in the Brownian sphere, a novel geometric insight into its structure.

Findings

No points are $k$-hubs for $k eq 2$ in the Brownian sphere

The Brownian sphere lacks complex branching points in geodesic structure

Geodesic concatenation properties are constrained in the Brownian sphere

Abstract

A point of a metric space is called a $k$-hub if it is the endpoint of exactly $k$ disjoint geodesics, and that the concatenation of any two of these paths is still a geodesic. We prove that in the Brownian sphere, there is no $k$-hub for $k\geq 3$.