Note on locality of volume growth rate of graphs

Abstract

In this note, we discuss locality of volume growth rate gr(G) and obtain the following results: (i) The volume growth rate is upper semi-continuous with respect to the local convergence, thus the locality holds when target graph is of volume growth rate 1. (ii) For any Cayley graph sequence {Gn}n=1∞ converging locally to regular tree Td(d≥3),limn→∞gr(Gn)=gr(Td). (iii) Given a finitely generated infinite group Γ=〈S|R〉 satisfying that there is a∈S of infinite order, and ∀b∈S,∃c∈〈S∖{a}|R〉 such that ba=ac. Then for Cayley graph sequence {Gn}n=1∞ of groups 〈S|R,an〉, which converges locally to Cayley graph G of Γ,gr(Gn)=gr(G).