Small growth rates of free groups

Abstract

We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property. Using this property, we determine the second and third smallest growth rates of the rank-$d$ free group for $d\ge 2$. We also give a new lower bound for the smallest growth rate of the genus-$g$ surface group for $g\ge 2$, as well as a lower bound for the growth rate associated with a one-relator presentation.

Disclosure

“Section 6. 1.5. The use of AI in this paper. Lemma 1.3 is one of the key steps in the proof of Theorem 1.1. The formulation of this lemma and the main ideas of its proof were developed through our extensive discussions with the AI model, ChatGPT 5.5 Pro by OpenAI. The authors then refined and generalized the arguments to ensure rigor and completeness, providing additional details and computations. We emphasize that the paper was written entirely by the authors.”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T18:58:30.668131Z

Structural counts