[论文解读] Density of growth rates of subgroups of a free group -- an alternative proof

Ádám Timár|arXiv (Cornell University)|Jan 18, 2026

Geometric and Algebraic Topology被引用 0

一句话总结

论文提供了一种简短、初级的替代证明，证明自由群 F_r 的有限生成子群的增长率在区间 [1, 2r-1] 上是稠密的。通过构造强周期树并将增长与非回退谱联系起来实现此结果。

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 $[1,2r-1]$.

研究动机与目标

• Motivate the problem of determining which growth rates occur for subgroups of a free group F_r (r ≥ 2).
• Reduce the problem to density of leading non-backtracking eigenvalues of finite graphs with degrees in {2,...,2r}.
• Provide an elementary construction showing the density of growth rates via universal covers of finite graphs.
• Relate subgroup growth to Perron eigenvalues of non-backtracking matrices and to universal covers of graphs.
• Offer a self-contained alternative to the Louvaris-Wise-Yehuda approach.

提出的方法

• Map finitely generated subgroups of F_r to Schreier graphs by choosing a basepoint, making the graph 2r-regular, and labeling edges by generators.
• Use the non-backtracking matrix B_G of the graph G to connect growth rate to its leading eigenvalue λ_1(B_G).
• Show that density of growth rates in [1,2r-1] follows from density of λ_1(B_G) values for finite graphs with degrees in {2,...,2r}.
• Prove a lemma: subdividing edges along a well-spaced set F of edges increases growth rate by at least a factor gr(T)^{n/(n+1)} while keeping strong periodicity.
• Construct strongly periodic trees by edge subdivisions and colorings to interpolate growth rates between (d-1)^{1/K} and (d-1)^{1/(2K)}.

实验结果

研究问题

• RQ1 What are the possible growth rates of finitely generated subgroups of a free group F_r?
• RQ2 Is the set of these growth rates dense in [1, 2r-1]?
• RQ3 Can one realize any target growth rate in (1, 2r-1) via strongly periodic trees or universal covers?
• RQ4 How does the non-backtracking spectrum relate to subgroup growth through Schreier graphs?

主要发现

• The growth rates of finitely generated subgroups of F_r are dense in [1, 2r-1].
• The Perron eigenvalue of the non-backtracking matrix of a Schreier graph captures the subgroup’s growth rate in F_r.
• A simple subdivision/edge-length manipulation preserves strong periodicity and controls growth rate, enabling interpolation between endpoints.
• A constructive scheme using universal covers and edge colorings yields a sequence of strongly periodic trees with growth rates approaching any target in the interval.
• The result provides an accessible alternative to probabilistic/complex arguments previously used to establish density.