Subgroups of a free group with every growth rate

TL;DR

This work proves that every value $\alpha$ in the interval $[1,2r-1]$ occurs as the exponential growth rate of some subgroup $H\le F_r$ (not necessarily finitely generated). The authors encode subgroups as immersed graphs in the bouquet $B_r$, relate growth to the non-backtracking matrix via $\omega(G)=\lambda_1(B_G)$ for finite graphs, and leverage density results to approximate $\alpha$ with spectra of finite graphs. By constructing an increasing sequence of finite subgraphs and glueing them along a long path, they use a limiting argument and the Collatz–Wielandt framework to realize $\omega_{F_r}(H)=\alpha$ for an infinite subgroup $H=\pi_1(G)$. A complementary discussion on the spectra of infinite graphs clarifies when the growth rate corresponds to an eigenvalue versus the approximate spectrum, connecting graph theory, operator theory, and geometric group theory. The results extend density phenomena to a constructive realization of all possible growth rates within the free-group setting.

Abstract

For every $α\in [1,2r-1]$, we show there exists a subgroup $H<F_r$ whose growth rate is $α$.

Figures (1)

• Figure 1: Construction of graph with prescribed growth.

Theorems & Definitions (25)

• Theorem 1.1: Main Theorem
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Remark
• Corollary 2.4
• proof : Proof of Theorem \ref{['thrm:labeling-infinite-graph']}
• Remark