Phase transitions in infinitely generated groups, and related problems in additive number theory

TL;DR

This paper investigates the growth functions of groups generated by infinite sets, revealing phase transition phenomena and providing a comprehensive classification of possible growth behaviors.

Contribution

It characterizes all possible growth functions for groups with infinite generating sets and describes a phase transition phenomenon, advancing understanding in group growth theory.

Findings

Classified all growth functions L_A(r) for infinite generating sets

Identified a phase transition phenomenon in group growth

Presented open problems in additive number theory

Abstract

Let A be an infinite set of generators for a group G, and let L_A(r) denote the number of elements of G whose word length with respect to A is exactly r. The purpose of this note is to determine all growth functions L_A(r) associated to infinite generating sets for groups, and to describe a phase transition phenomenon associated with infinite generating sets. A list of open problems is also.included.