自动群有理子集的Stallings折叠方法
Stallings foldings for rational subsets of automatic groups
中文总结 AI 辅助
本研究针对自动群的有理子集，在Kharlampovich等关于子群的工作基础上，提出基于Stallings折叠构造识别自动机的方法，针对子幺半群可检验构造完备性，结合小消去理论应用于亏格大于1的曲面群。
中文摘要 AI 辅助
设$G$为带有关联正则语言$L$的自动群，我们描述一种构造自动机的方法，该自动机可识别$G$中给定子幺半群或有理子集$K$的元素。该方法基于Kharlampovich、Miasnikov和Weil关于$K$为$G$子群情形的研究工作。只要$K$满足称为$L$-邻近性的特定凸性性质，经过足够多次迭代后我们的构造即可成功。我们展示了当$K$为子幺半群时如何检验构造是否完备，但对于$K$为有理子集的一般情形尚无此类检验方法。我们特别关注亏格$g>1$的曲面群$G$，其中$L$是标准生成元下的测地线字语言，我们利用小消去理论得到构造$G$的$L$-可识别子幺半群的方法。
英文摘要
Let $G$ be an automatic group with associated regular language $L$. We describe a procedure for constructing an automaton which recognises elements of a given submonoid or rational subset $K$ of $G$. This builds on work of Kharlampovich, Miasnikov and Weil, on the case where $K$ is a subgroup of $G$. Our construction succeeds, after sufficiently many iterations, whenever $K$ satisfies a certain convexity property, which we call $L$-proximity. We show how to test whether the construction is complete in the case that $K$ is a submonoid; we have no such test for the general case of a rational subset $K$. We focus particularly on the case of a surface group $G$ of genus $g>1$, where $L$ is the language of geodesic words in the standard generators. We use small cancellation theory to obtain a method for constructing $L$-recognisable submonoids of $G$.
