Growth of Primitive Elements in Free Groups

TL;DR

This paper characterizes the typical structure of primitive elements in free groups, showing their asymptotic letter distribution, and determines the exponential growth rate of primitive words.

Contribution

It provides a detailed description of generic primitive elements and establishes their growth rate, answering an open problem in combinatorial group theory.

Findings

A random primitive word contains each letter exactly once asymptotically almost surely.

The exponential growth rate of primitive words in $F_k$ for $k geq 2$ is $2k-3$.

The method applies to elements in proper free factors as well.

Abstract

In the free group $F_k$, an element is said to be primitive if it belongs to a free generating set. In this paper, we describe what a generic primitive element looks like. We prove that up to conjugation, a random primitive word of length $N$ contains one of the letters exactly once asymptotically almost surely (as $N \to \infty$). This also solves a question from the list `Open problems in combinatorial group theory' [Baumslag-Myasnikov-Shpilrain 02']. Let $p_{k,N}$ be the number of primitive words of length $N$ in $F_k$. We show that for $k \ge 3$, the exponential growth rate of $p_{k,N}$ is $2k-3$. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor.