(Random) Trees of Intermediate Volume Growth

ABSTRACT For every function that grows at least linearly and at most exponentially, if it is sufficiently well‐behaved, we can construct a tree of uniform volume growth , or more precisely, for constants that only depend on . Here denotes the ball of radius centered at a vertex . In particular, this yields examples of trees of uniform intermediate (i.e., super‐polynomial and sub‐exponential) growth. We use this construction to provide the first examples of unimodular random rooted trees of uniform intermediate growth, answering a question by Itai Benjamini. We find a peculiar change in structural properties for these trees at growth . Our results can be applied to obtain triangulations of for with various uniform growth behaviors, as well as Riemannian metrics on for the same wide range of growth behaviors.