从代数正交性到分层双曲群中的RAAG嵌入障碍
From algebraic orthogonality to RAAG embedding obstructions in hierarchically hyperbolic groups
AI总结：
该研究针对两类分层双曲群结构引入扩展核心图，证明直角阿廷群嵌入可经中间RAAG分解，给出RAAG嵌入障碍与判别准则，并建立相关不变性结果。
AI中文摘要：
我们针对两类以紧特殊群和映射类群为模型的分层双曲群（HHG）结构，引入了记录极小无界区域及其轴向方向的“扩展核心图”。我们的主要结构结果表明：每个直角阿廷群（RAAG）的嵌入，在将其标准生成元替换为正幂次后，可通过由适当支撑的轴向元生成的中间RAAG进行分解；对于以紧特殊群为模型的这类结构，该中间RAAG是拟等距嵌入的。我们证明其扩张图可嵌入扩展核心图，这给出了Kim-Koberda型的RAAG嵌入障碍，且当秩不超过2时，可得到完整的嵌入判别准则。对于映射类群上的标准HHG结构，扩展核心图是本质曲线的不交图；而对于RAAG上的自然丰富族结构，它可恢复扩张图。我们还建立了有限直积和标准相对双曲构造下的不变性结果。
英文摘要：
We introduce the $\textit{expanded core graph}$, which records minimal unbounded domains and their axial directions, for two classes of hierarchically hyperbolic group structures modeled on compact special groups and mapping class groups. Our main structural result shows that every embedding of a right-angled Artin group, after replacing its standard generators by positive powers, factors through an intermediate RAAG generated by suitably supported axial elements; for the class modeled on compact special groups, this intermediate RAAG is quasi-isometrically embedded. We show that its extension graph embeds into the expanded core graph. This yields a Kim--Koberda-type obstruction to RAAG embeddings and a complete embedding criterion when the rank is at most two. For the standard HHG structure on a mapping class group, the expanded core graph is the disjointness graph of essential curves, while for natural rich-family structures on a RAAG it recovers the extension graph. We also establish permanence results under finite direct products and the standard relatively hyperbolic construction.
