A Unimodular Random Graph With Large Upper Growth and No Growth

ABSTRACT We construct a unimodular random rooted graph with maximal degree and upper growth rate , which does not have a growth rate. Abért, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least , then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non‐hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Abért that unimodular random Riemannian surfaces of bounded negative curvature always have growth.