Subgroups of a free group with every growth rate

TL;DR

This paper demonstrates that for any growth rate within a specific range, there exists a subgroup of a free group that exhibits exactly that growth rate, expanding understanding of subgroup growth behaviors.

Contribution

It constructs subgroups of free groups with arbitrary growth rates in a continuous interval, a novel achievement in subgroup growth theory.

Findings

Existence of subgroups with all growth rates in [1, 2r-1]

Explicit construction methods for such subgroups

Extension of growth rate classification in free groups

Abstract

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