The subgroup growth spectrum of virtually free groups

TL;DR

This paper proves that the growth coefficient of virtually free groups is always rational, can take any rational value, and provides an algorithm to compute it.

Contribution

It establishes the rationality of the growth coefficient for virtually free groups, characterizes all possible values, and introduces a computational method.

Findings

Growth coefficient of virtually free groups is always rational.

Every rational number can be realized as a growth coefficient.

An algorithm to compute the growth coefficient is provided.

Abstract

For a finitely generated group $\Gamma$ denote by $\mu(\Gamma)$ the growth coefficient of $\Gamma$, that is, the infimum over all real numbers $d$ such that $s_n(\Gamma)<n!^d$. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group. Moreover, we describe an algorithm to compute $\mu$.