The virtually-free cogrowth conjecture

Let $\Gamma=\langle S\mid R\rangle$ be a group with finite generating set $S$, and let

$$\Psi_{\Gamma,S}(z)=\sum_{n=0}^{\infty}\psi_nz^n$$

be its cogrowth series, where $\psi_n$ counts words of length $n$ in the alphabet $S\cup S^{-1}$ that represent the identity in $\Gamma$. Virtually-free cogrowth conjecture. The cogrowth series $\Psi_{\Gamma,S}(z)$ is algebraic if and only if $\Gamma$ is virtually free. The reverse implication follows from theorems of Muller–Schupp and Chomsky–Schützenberger, while the forward implication remains open; the conjecture is a cogrowth formulation of a longstanding question often expressed using Green's functions.