论文标题
死胡同和完整增长系列的合理性
Dead ends and rationality of complete growth series
论文作者
论文摘要
有限生成的组的完整增长系列由$ \ sum_ {n \ ge 0} a_ns^n $给出，其中$ a_n $是组半段中长度$ n $的元素的总和。我们研究了此类系列的$ \ Mathbb ng $ - 理性和$ \ Mathbb ng $ -Algebraicity。我们表明，任意深度的死胡同是$ \ Mathbb ng $ - 理性的障碍。对于$ 3 $二维的Heisenberg Group $ H_3（\ Mathbb Z）$，我们证明完整系列不是任何生成集的$ \ Mathbb ng $ -Algebraic。死胡同还用于表明，对于特定生成集的，高级海森堡小组的完整增长系列不是$ \ mathbb ng $合理。使用此阻塞的更通用的版本，我们证明了一些Lamplighter组的完整增长系列也不是$ \ Mathbb ng $合理。
The complete growth series of a finitely generated group is given by $\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the group semiring. We study the $\mathbb NG$-rationality and $\mathbb NG$-algebraicity of such series. We show that having dead ends of arbitrarily large depths is an obstruction to $\mathbb NG$-rationality. In the case of the $3$-dimensional Heisenberg group $H_3(\mathbb Z)$, we prove that the complete series is not $\mathbb NG$-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not $\mathbb NG$-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not $\mathbb NG$-rational either.
