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The subgroup growth spectrum of virtually free groups

For a finitely generated group $Γ$ denote by $μ(Γ)$ the growth coefficient of $Γ$, that is, the infimum over all real numbers $d$ such that $s_n(Γ)<n!^d$. 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 $μ$.