Density of growth rates of subgroups of a free group -- an alternative proof

TL;DR

This work addresses which growth rates $\omega(H)$ can occur for finitely generated subgroups of a free group $F_r$, proving that these rates are dense in the interval $[1,2r-1]$. It offers an elementary approach by linking subgroup growth to the Perron eigenvalues of non-backtracking matrices on Schreier graphs, equivalently to growth rates of universal covers (strongly periodic trees). A central technical component is a subdivision lemma that controllably decreases growth through edge subdivisions, enabling a constructive interpolation between growth values. The result provides a conceptually transparent alternative to the original probabilistic graph-construction proof, highlighting the connection between growth, non-backtracking spectral theory, and graph immersions in the free-group setting.

Abstract

We give an alternative proof to the theorem recently proved by Louvaris, Wise and Yehuda, that the growth rates of finitely generated subgroups of $F_r$ are dense in $[1,2r-1]$.

Theorems & Definitions (5)

• Theorem 1: Louvaris-Wise-Yehuda, LWY2
• Theorem 2: Louvaris-Wise-Yehuda, LWY2
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['graph']}