发表机构
Kyungpook National University; Pusan National University（庆北国立大学; 釜山国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在图辫群的万有覆盖上构造显式因子系统，建立分层双曲群结构，给出自由阿贝尔子群最大秩的有限公式，并证明所有RAAG均可无扭曲嵌入，且对图2-辫群给出二分图限制与图论判据。
AI 中文摘要
对于连通有限图$\mathsf{\Gamma}$上$n$个粒子的无序离散构型空间$\mathrm{UD}_n(\mathsf{\Gamma})$，我们在其万有覆盖上构造了一个显式的因子系统。其因子由合法对（即配备粒子分布的子图）编码。在所得的分层双曲群（HHG）结构中，嵌套、正交和乘积区域可通过构型空间几何得到显式描述，并且我们证明该结构满足构造和阻碍与直角阿廷群（RAAGs）同构的子群所需的附加性质。利用充分细分的模型，我们将此分层结构应用于图辫群。我们给出了自由阿贝尔子群最大秩的有限组合公式，并证明每个RAAG都作为某个图辫群的无扭曲子群出现。对于图$2$-辫群，我们得到更强的限制：每个RAAG子群具有二分定义图，且嵌入问题可由分层结构的扩展核图中的诱导子图条件来刻画。对于由四顶点路径定义的RAAG，该条件等价于底层图上的有限图论判据。
英文摘要
For the unordered discrete configuration space $\mathrm{UD}_n(\mathsfΓ)$ of $n$ particles on a connected finite graph $\mathsfΓ$, we construct an explicit factor system on its universal cover. Its factors are encoded by legal pairs, namely subgraphs equipped with particle distributions. The nesting, orthogonality, and product regions in the resulting hierarchically hyperbolic group (HHG) structure admit explicit descriptions in terms of configuration-space geometry, and we show that this structure satisfies the additional properties needed for constructing and obstructing subgroups isomorphic to right-angled Artin groups (RAAGs). Using sufficiently subdivided models, we apply this hierarchy to graph braid groups. We give a finite combinatorial formula for the maximal rank of a free abelian subgroup and show that every RAAG occurs as an undistorted subgroup of some graph braid group. For graph $2$-braid groups, we obtain stronger restrictions: every RAAG subgroup has bipartite defining graph, and the embedding problem is characterized by an induced-subgraph condition in the expanded core graph of the hierarchy. For the RAAG defined by the four-vertex path, this condition is equivalent to a finite graphical criterion on the underlying graph.
Comments48 pages, 16 figures. Comments are welcome!
