The branching number of intermediate growth trees

TL;DR

This paper introduces the intermediate branching number (IBN) for intermediate growth trees, establishing it as a critical threshold for random processes and providing algorithms to construct trees with large IBN, aiding the study of intermediate growth groups.

Contribution

It defines the IBN, analyzes its properties, and develops an algorithm to construct spherically symmetric trees with large IBN within permutation wreath products.

Findings

IBN is the critical threshold for certain random processes on trees.

Algorithms can construct trees with large IBN in permutation wreath products.

First tight bounds for firefighter problem on intermediate growth groups.

Abstract

We introduce an "intermediate branching number"(IBN) which captures the branching of intermediate growth trees, similar in spirit to the well-studied branching number of exponential growth trees. We show that the IBN is the critical threshold for several random processes on trees, and analyze the IBN on some examples of interest. Our main result is an algorithm to find spherically symmetric trees with large IBN inside some permutation wreath products. We demonstrate the usefulness of these trees to the study of intermediate growth groups by using them to get the first tight bounds for the firefighter problem on some inetrmediate growth groups.