A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs

TL;DR

This paper establishes a coarse version of Halin's Grid Theorem for certain infinite graphs, showing they contain the half-grid as an asymptotic minor, with applications to quasi-transitive, locally finite graphs and Cayley graphs.

Contribution

It proves a coarse Halin Grid Theorem for a broad class of graphs, resolving a conjecture and applying to quasi-transitive, locally finite graphs including Cayley graphs.

Findings

Every one-ended, locally finite graph with infinitely many rays contains the half-grid as an asymptotic minor.

The theorem applies to all one-ended, quasi-transitive, locally finite graphs.

It includes all locally finite Cayley graphs of one-ended finitely generated groups.

Abstract

We prove a coarse version of Halin's Grid Theorem: Every one-ended, locally finite graph that contains the disjoint union of infinitely many rays as an asymptotic minor also contains the half-grid as an asymptotic minor. More generally, we show that the same holds for arbitrary (not necessarily one-ended or locally finite) graphs under additional, necessary assumptions on the minor-models of the infinite rays. This resolves a conjecture of Georgakopoulos and Papasoglu. As an application, we show that every one-ended, quasi-transitive, locally finite graph contains the half-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of one-ended finitely generated groups and solves a problem of Georgakopoulos and Papasoglu.