Growth Rates of Finitely Generated Subgroups

• Growth rates of finitely generated subgroups describe the asymptotic count of elements within word metric balls, highlighting key structural properties in various groups.
• Graph-theoretic and spectral techniques, such as non-backtracking operators on Schreier graphs, are used to establish density theorems and analyze growth spectra.
• The analysis uncovers contrasting phenomena: free groups exhibit a dense spectrum of growth rates while hyperbolic groups display a well-ordered and rigid growth structure.

A finitely generated subgroup, $H$, of a group $G$ is characterized by the asymptotic behavior of the number of its elements within balls of increasing radius in the ambient group's word metric. The study of growth rates of such subgroups elucidates deep structural properties, ranging from combinatorics and spectral theory to rigidity phenomena across group-theoretic landscapes. For free groups and, more generally, for groups with rich geometric structure (e.g., hyperbolic, acylindrically hyperbolic, virtually abelian), the growth rate spectrum captures both universal and sharply contrasting features.

1. Definitions and Foundational Results

Let $F_r$ be a free group of rank $r \geq 2$ with free generating set $S$. The word length $|g|_S$ of $g \in F_r$ is the minimal number of generators from $S \cup S^{–1}$ needed to express $g$. For any subgroup $H \leq F_r$, the associated growth function is

$\gamma_H(n) = |\{ h \in H : |h|_S \leq n \} |$

and its (exponential) growth rate is

$$\omega(H) = \limsup_{n \to \infty} \gamma_H(n)^{1/n}.$$

Fekete’s Lemma ensures this limsup is a limit, and one always has $1 \leq \omega(H) \leq 2r-1 = \omega(F_r)$. The trivial subgroup and infinite cyclic (undistorted) subgroups realize $\omega(H) = 1$, while $F_r$ itself has the maximal rate.

Equivalently, $\omega(H)$ is the Perron–Frobenius eigenvalue of the non-backtracking (Hashimoto) operator on the Schreier (or Stallings) graph associated to $H$ (Timár, 18 Jan 2026).

2. Density and Structure of Growth Rates in Free Groups

A central result established by Louvaris, Wise, and Yehuda, with an alternative proof by (Timár, 18 Jan 2026), is the density theorem:

Theorem. For any real $\alpha$ with $1 \leq \alpha \leq 2r-1$ and any $\varepsilon > 0$, there exists a finitely generated subgroup $H < F_r$ such that

$|\omega(H) - \alpha| < \varepsilon.$

The proof leverages reductions to finite graphs and the Hashimoto operator: any finite connected graph $G$ of degree in $\{2, \dots, 2r\}$ can be made into a Schreier graph for some $H < F_r$, and closed non-backtracking walks relate bijectively to elements of $H$ of a given length. As a consequence, the spectrum of possible growth rates, as realized by finitely generated subgroups of $F_r$, is dense in $[1,2r-1]$—every rate in this interval can be approximated arbitrarily closely by such subgroups (Timár, 18 Jan 2026).

3. Methodologies: Combinatorial and Spectral Techniques

The density theorem's proof is anchored in:

• Graph-theoretic reduction: Any finite graph with degree constraints is realized as a Schreier graph, and its non-backtracking walk operator's spectral radius coincides with the exponential growth rate.
• Strongly periodic trees: These are universal covers of such finite graphs and have growth rates matching the corresponding Hashimoto operator's dominant eigenvalue.
• Sparse edge subdivisions: By performing controlled edge subdivisions (parameterized by coloring and girth), the set of achievable growth rates is shown to densely interpolate between the bounds $(2r-1)^{1/K}$ and $(2r-1)^{1/(2K)}$, for large $K$.

By assembling these ingredients, one constructs, for any $\alpha \in [1,2r-1]$, a graph and thus a subgroup whose growth is arbitrarily close to $\alpha$ (Timár, 18 Jan 2026).

4. Comparative Analysis in Hyperbolic and Related Groups

For hyperbolic groups in general, the situation is sharply distinct: the set of growth rates of finitely generated subgroups (with respect to all generating sets) is always well-ordered—there can be no infinite descending chains. In the case of limit groups (including free and surface groups), this order type is $\omega^\omega$ (Fujiwara et al., 2020). For each $r > 1$, only finitely many non-isomorphic subgroup/generating set pairs can realize a given growth rate, marking a profound rigidity compared to the dense interval for $F_r$.

In contrast, for finitely generated virtually abelian groups, the growth (as an integer sequence) is "rational": the growth series of any finitely generated subgroup is a rational function in the formal variable $z$ (Evetts, 2018).

Product-set growth phenomena further distinguish mapping class groups and right-angled Artin groups: in these groups, a strict dichotomy prevails—finitely generated subgroups either have (virtual) infinite center or satisfy a uniform exponential lower bound on their setwise product growth, with explicit constants (Kerr, 2021).

5. Subgroup Growth, Distortion, and Associated Spectra

The relative growth and distortion functions intimately connect to subgroup embedding properties.

• For non-locally finite subgroups $H$ in finitely generated $G$, the relative growth $g_{H,G}(n)$ is, up to equivalence, a superadditive function, and every such function (up to exponential) can be realized as the growth rate of a cyclic subgroup embedded in some solvable group (Davis et al., 2012).
• In free groups, characteristic subgroup growth is of type $n^{\log n}$, strictly intermediate between polynomial and exponential, with no distinction between ranks $r \geq 2$ as previously conjectured (Hanany et al., 5 Oct 2025).

Groups with intermediate growth—between polynomial and exponential—arise in the context of topological full groups and inverse semigroups of bounded type. Under finiteness of incompressible elements, every finitely generated subgroup of such a full group is periodic and exhibits growth bounded above by $\exp(R^\alpha)$ for some $\alpha < 1$ (Kuang, 2024), supplying explicit examples in the intermediate regime.

6. Open Questions and Future Directions

Significant open problems remain:

• What is the precise spectrum of growth rates for normal subgroups of $F_r$?
• Are the analogous density results for growth rates valid for subgroups of hyperbolic or relatively hyperbolic groups, or do gaps always exist (as is the case for the possible spectral radii)?
• For groups beyond free or hyperbolic classes, can the techniques, especially those involving combinatorial tree constructions and non-backtracking operators, be extended to determine quantitative density or well-ordering properties?
• Classification of the spectra and growth types for subgroups in groups defined by dynamic or topological full group actions remains an active area (Kuang, 2024).

7. Broader Significance and Comparative Summary

The spectrum of growth rates of finitely generated subgroups serves as a critical invariant, varying dramatically depending on the group class. In free groups, the interval $[1,2r-1]$ is densely filled, exhibiting a sharp "no-gap" phenomenon (Timár, 18 Jan 2026). In Gromov-hyperbolic groups, the structure is discretely well-ordered, with additional rigidity: only finitely many non-isomorphic subgroups can realize any given rate (Fujiwara et al., 2020). In virtually abelian groups, growth functions are algebraic (rational), while group actions arising from bounded-type inverse semigroups can produce a variety of growth behaviors, including explicit intermediate cases (Kuang, 2024).

This landscape underscores the deep interplay between geometric, spectral, and combinatorial aspects that govern the possible growth rates of finitely generated subgroups across group-theoretic contexts.