Possible heights of graph transformation groups

TL;DR

This paper characterizes the possible heights of transformation groups on topological graphs, showing that for any finite height p, there exists a graph with exactly those heights plus infinity, and that these sets are complete for graphs with height p.

Contribution

The paper establishes a comprehensive classification of all possible heights of transformation groups on topological graphs, including existence and characterization results.

Findings

For each finite p, a topological graph with all heights from p to infinity exists.

The set of all possible heights for a given graph with height p is exactly {p, p+1, p+2, ... , +∞}.

The classification applies universally to all topological graphs with a given height p.

Abstract

In the following text we prove that for all finite $p\geq0$ there exists a topological graph $X$ such that $\{p,p+1,p+2,\ldots\}\cup\{+\infty\}$ is the collection of all possible heights for transformation groups with phase space $X$. Moreover for all topological graph $X$ with $p$ as height of transformation group $(Homeo(X),X)$, $\{p,p+1,p+2,\ldots\}\cup\{+\infty\}$ again is the collection of all possible heights for transformation groups with phase space $X$.