Metric dimensions of minor excluded graphs and minor exclusion in groups

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. 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, like Z3; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.