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arXiv · 2607.17444

Small growth rates of free groups

Koji Fujiwara·Sang-hyun Kim·Ryokichi Tanaka
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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.

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Koji Fujiwara, Sang-hyun Kim, Ryokichi Tanaka. 2026-07-20. Small growth rates of free groups. https://arxiv.org/abs/2607.17444

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