Metric dimensions of minor excluded graphs and minor exclusion in groups

TL;DR

This paper explores the properties of minor excluded graphs, especially Cayley graphs of groups, revealing how minor exclusion relates to group structure, generating sets, and graph dimensions.

Contribution

It demonstrates that minor exclusion varies with generating sets, characterizes groups with minor excluded Cayley graphs, and establishes preservation under free products.

Findings

Minor excluded graphs have finite Assouad-Nagata dimension.

Minor exclusion depends on the generating set, not just the group.

Virtually free groups are minor excluded for all generating sets.

Abstract

An infinite graph G is minor excluded if there is a finite graph that is not a minor of G. We prove that minor excluded graphs have finite Assouad-Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.