A metric between quasi-isometric trees

TL;DR

This paper introduces a new metric for comparing rooted geodesically complete simplicial trees based on their boundary ultrametric spaces, providing a way to quantify their quasi-isometry and branching structure.

Contribution

It defines a novel boundary-based metric for trees that captures their quasi-isometric and branching properties, extending the understanding of ultrametric space boundaries.

Findings

The metric characterizes the branching of ultrametric spaces.

It measures the distance between trees in the same quasi-isometry class.

The metric quantifies how close trees are to being rooted isometric.

Abstract

It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete \br-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that, when the ultrametric spaces are the corresponding end spaces, this map defines a metric between rooted geodesically complete simplicial trees with minimal vertex degree 3 in the same quasi-isometry class. Moreover, this metric measures how far are the trees from being rooted isometric.