Fat minors cannot be thinned (by quasi-isometries)

TL;DR

The paper disproves a conjecture that certain length spaces or graphs with no fat minors are quasi-isometric to minor-free graphs, providing counterexamples and a weakened positive result for 3-fat minors.

Contribution

It provides the first counterexamples to the conjecture and establishes a weakened quasi-isometry result for 3-fat minors.

Findings

Counterexamples to the conjecture exist.

Graphs with no $K$-fat $H$ minor are quasi-isometric to graphs with no 3-fat $H$ minor.

The original conjecture does not hold in general.

Abstract

We disprove the conjecture of Georgakopoulos and Papasoglu that a length space (or graph) with no $K$-fat $H$ minor is quasi-isometric to a graph with no $H$ minor. Our counterexample is furthermore not quasi-isometric to a graph with no 2-fat $H$ minor or a length space with no $H$ minor. On the other hand, we show that the following weakening holds: any graph with no $K$-fat $H$ minor is quasi-isometric to a graph with no $3$-fat $H$ minor.