Numerical upper bounds on growth of automata groups

TL;DR

This paper introduces an algorithmic method to compute numerical upper bounds on the growth exponents of automata groups, aiding in understanding their growth behavior and discovering new examples of intermediate growth groups.

Contribution

It provides a novel algorithmic procedure implemented in GAP to estimate growth bounds of automata groups, including known and new examples of intermediate growth.

Findings

Retrieved optimal bounds for the Grigorchuk group

Improved bounds on other automata groups

Discovered new automata groups with intermediate growth

Abstract

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automata groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automata groups and permitted us to discover several new examples of automata groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.