Dead ends and rationality of complete growth series (Q6608720)

Let \(G\) be a group generated by a finite, symmetric set \(S\). Consider the sequence \((a_{n})\) that counts the number of elements at distance \(n\) from the identity, in the word metric relative to \(S\). The growth of this sequence is usually studied through the numerical growth series of \(G\)\N\[\NG_{\mathrm{num}}(s) = \sum_{n \ge 0} a_{n} s^{n} \in \mathbb{N}[[s]].\N\]\NThe paper under review is concerned with the complete growth series of \(G\)\N\[\NG(s) = \sum_{n \ge 0} A_{n} s^{n},\N\]\Nwhere \(A_{n}\) is the sum of the elements of \(G\) at distance \(n\) from the identity. Thus \(G(s)\) lies in \(\mathbb{N} G[[s]]\), where \(\mathbb{N} G\) is the group semiring. \N\NFor a series with coefficients in a semiring \(R\), there are concepts of \(R\)-rationality and \(R\)-algebraicity. Both concepts require the series to satisfy a certain system of equations, which are linear in the first case and polynomial in the second case. \N\N\textit{F. Liardet} [Croissance dans les groupes virtuellement abéliens. Genève: Université de Genève (PhD thesis) (1996)] has proved that the complete growth series of a virtually abelian group is \(\mathbb{N}G\)-rational for any finite generating set. \textit{R. Grigorchuk} and \textit{T. Nagnibeda} [Invent. Math. 130, No. 1, 159--188 (1997; Zbl 0880.20024)] have proved that the complete growth series of a Gromov hyperbolic group is \(\mathbb{N}G\)-rational for any finite generating set. \N\NThere are various implications for a group \(G\) between the concepts of \(\mathbb{N}\)-rationality, \(\mathbb{N}G\)-algebraicity, \(\mathbb{N}G\)-rationality and \(\mathbb{N}G\)-algebraicity, such as that \(R\)-rationality implies \(R\)-algebraicity. In the paper under review, the authors provide examples to settle in the negative the following question: for a group \(G\), does an \(\mathbb{N}\)-property imply the corresponding \(\mathbb{N}G\)-property? For instance, the Heisenberg group \(G = H_{3}(\mathbb{Z})\) is \(\mathbb{N}\)-rational but not \(\mathbb{N}G\)-rational. \N\NWe refer to the introduction of the paper, which is clearly written and informative, for further details, in particular concerning the role played by the so-called dead ends.