Foldings, graphs of groups and the membership problem

Membership Problem in groups acting freely on Z^n-trees

Groups acting freely on Z^n-trees (Z^n-free groups) play a key role in the study of non-archimedean group actions. Following Stallings' ideas, we develop graph-theoretic techniques to investigate subgroup structure of Z^n-free groups. As an immediate application of the presented method, we give an effective solution to the Uniform Membership Problem and the Power Problem in Z^n-free groups.

关于扭曲右角Artin群的子群可分性与成员判定问题

On Subgroup Separability and Membership Problems in Twisted Right-Angled Artin Groups

本文仅通过定义混合图刻画了扭曲右角Artin群的子群可分性，推广了经典结果，并证明了凝聚群的子群成员判定问题可判定，同时解决了特定锥图族对应的有理与子半群成员判定问题。

Folding-like techniques for CAT(0) cube complexes

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fund…

辫子群与 Artin 群中的成员问题

Membership problems in braid groups and Artin groups

本文依据定义图的禁用诱导子图分类了多种成员问题可判定的 Artin 群，并证明辫子群 $\mathbf{B}_n$ 中这些问题可判定当且仅当 $n\leq3$。

折叠画廊与矩图

Folded galleries and moment graphs

本文通过建立与球面Coxeter群的Bruhat矩图的联系，刻画了仿射Coxeter复形中极小alcove-to-alcove画廊的正向折叠模式，并利用该图确定了正向折叠画廊末端alcove的球面方向。

Finite automata for Schreier graphs of virtually free groups

The Stallings construction for finitely generated subgroups of free groups is generalized by introducing the concept of Stallings section, which allows an eficient computation of the core of a Schreier graph based on edge folding. It is proved that those groups admitting Stallings sections are precisely finitely generated virtually free groups, through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.

关于有界自动群中ET0L子群成员问题

On the ET0L subgroup membership problem in bounded automata groups

本文研究了有界自动群中无限射线稳定子群的成员问题，证明其形成可构造的ET0L语言，并给出了生成函数的递归公式。

Artin群的子半群与有理子集成员资格问题

The submonoid and rational subset membership problems for Artin groups

本文证明了在Artin群中子半群与有理子集成员资格问题等价，并利用定义图刻画了其可判定性，推广了直角Artin群的相关结论。

$\mathbb{Z}$的类平移作用、子群成员问题与有效子转移的Medvedev度

Translation-like actions by $\mathbb{Z}$, the subgroup membership problem, and Medvedev degrees of effective subshifts

研究证明无限局部有限连通图存在Z的类平移作用（单/两端图可传递且d(v,v*1)≤3，强化Seward定理），还证明具可判定字问题的有限生成无限群有可计算类平移作用且满足可判定轨道成员问题，应用得此类群的有效子转移可达所有Π₁⁰Medvedev度，扩展Miller对Zᵈ的分类。