Growth and Language Complexity of Potentially Positive Elements of Free Groups

TL;DR

This paper develops automata-based methods to analyze and bound the growth of potentially positive elements in free groups, providing tight bounds for the case of two generators and exploring automata recognition limitations.

Contribution

It introduces automata techniques to study potentially positive words in free groups and establishes tight asymptotic growth bounds for groups with 2 to 7 generators.

Findings

Tight growth bounds for F_2 case.

Automata can analyze properties of potentially positive words.

Certain automata cannot recognize all potentially positive elements.

Abstract

A word in a free group is called ``potentially positive'' if it is automorphic to an element which is written with only positive exponents. We will develop automata to analyze properties of potentially positive words. We will use these to give new bounds on the asymptotic growth of potentially positive elements in free groups of 2 to 7 generators. We prove the bounds for $F_2$ are tight, giving the growth function up to a constant multiplier. We use the same tools to show that certain restricted automata cannot recognize the set of potentially positive elements.