A unimodular random graph with large upper growth and no growth

TL;DR

This paper constructs a unimodular random graph with maximal degree d and upper growth rate d-1 that lacks a growth rate, providing a counterexample to previous conjectures and expanding understanding of growth properties in such graphs.

Contribution

It presents the first known example of a unimodular random graph with maximal degree d and upper growth rate d-1 that does not have a growth rate, challenging existing beliefs.

Findings

Constructed a unimodular random graph with no growth rate.

Provided a non-hyperfinite example of such a graph.

Countered previous conjectures about growth in unimodular graphs.

Abstract

We construct a unimodular random rooted graph with maximal degree $d\geq 3$ and upper growth rate $d-1$, which does not have a growth rate. Ab\'ert, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least $\sqrt{d-1}$, 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 Ab\'ert that unimodular Riemannian surfaces of bounded negative curvature always have growth.