On Infinitely Many Siblings for Locally Finite Trees with Parabolic Embeddings

TL;DR

This paper investigates the structure of locally finite trees with parabolic self-embeddings, showing they are mutually embeddable with infinitely many non-isomorphic trees unless they are a one-way infinite path.

Contribution

It proves that such trees have infinitely many siblings unless they are a one-way infinite path, clarifying the structure of trees with parabolic embeddings.

Findings

Locally finite trees with parabolic embeddings have infinitely many non-isomorphic siblings.

The exception is the one-way infinite path, which does not have infinitely many siblings.

Properties by Bonato-Tardif and Tyomkyn hold for trees without hyperbolic self-embeddings.

Abstract

Parabolic (resp. hyperbolic) self-embeddings of trees are those which do not fix a non-empty finite subtree and preserve precisely one (resp. two) end(s). We prove that a locally finite tree having a parabolic self-embedding is mutually embeddable with infinitely many pairwise non-isomorphic trees, unless the tree is a one-way infinite path. As a result, we conclude that two important properties identified by Bonato-Tardif and Tyomkyn hold for locally finite trees not having any hyperbolic self-embedding.